Theorem eqfnfv2f 5985

Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5981 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Hypotheses

Ref	Expression
eqfnfv2f.1
eqfnfv2f.2

Assertion

Ref	Expression
eqfnfv2f

Distinct variable group:   ,

Proof of Theorem eqfnfv2f

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqfnfv 5981	. 2
2		eqfnfv2f.1	. . . . 5
3		nfcv 2619	. . . . 5
4	2, 3	nffv 5878	. . . 4
5		eqfnfv2f.2	. . . . 5
6	5, 3	nffv 5878	. . . 4
7	4, 6	nfeq 2630	. . 3
8		nfv 1707	. . 3
9		fveq2 5871	. . . 4
10		fveq2 5871	. . . 4
11	9, 10	eqeq12d 2479	. . 3
12	7, 8, 11	cbvral 3080	. 2
13	1, 12	syl6bb 261	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  F/_wnfc 2605  A.wral 2807  Fnwfn 5588  `cfv 5593

This theorem is referenced by:  aacllem  33216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601