Is it not neccessary to push EAX in this code? [duplicate]

Question

I am new to assembly, and when I studied on the internet, I found the following code, but I don't know why push eax is not written before mov eax, [ebp+8].

; char* removerng(char* s, char a, char b)
; removes (in place) characters from a to b in string s
    section .text
    global  removerng
removerng:
    ; prologue
    push    ebp
    mov     ebp, esp
    push    ebx ;<- ebx instead of eax?
    push    esi
    push    edi

    ; body <- we don't need eax?
    mov eax, [ebp+8]; load string into return address
    mov esi, [ebp+8]; last char checked
    mov edi, [ebp+8]; last char placed

loop:
    mov bh, [esi]; current char
    inc esi
    test    bh, bh; test for end of string
    jz  fin
    cmp bh, [ebp+12] 
    jl  keep
    cmp bh, [ebp+16]
    jg  keep
    jmp loop

keep:
    mov [edi], bh
    inc edi
    jmp loop

fin:
    mov [edi], bh

    ;epilogue
    pop edi
    pop esi
    pop ebx
    mov esp, ebp
    pop ebp
    ret

Does this mean we can use both push ebx and push eax Could any one help me?