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Theorem eqfnfv2f 5823
 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 5819 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:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem eqfnfv2f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqfnfv 5819 . 2
2 eqfnfv2f.1 . . . . 5
3 nfcv 2571 . . . . 5
42, 3nffv 5727 . . . 4
5 eqfnfv2f.2 . . . . 5
65, 3nffv 5727 . . . 4
74, 6nfeq 2578 . . 3
8 nfv 1629 . . 3
9 fveq2 5720 . . . 4
10 fveq2 5720 . . . 4
119, 10eqeq12d 2449 . . 3
127, 8, 11cbvral 2920 . 2
131, 12syl6bb 253 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652  wnfc 2558  wral 2697   wfn 5441  cfv 5446 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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