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Theorem eqfnfv2f 5440
 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 5436 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 5436 . 2
2 eqfnfv2f.1 . . . . 5
3 nfcv 2235 . . . . 5
42, 3nffv 5350 . . . 4
5 eqfnfv2f.2 . . . . 5
65, 3nffv 5350 . . . 4
74, 6nfeq 2243 . . 3
8 nfv 1473 . . 3
9 fveq2 5340 . . . 4
10 fveq2 5340 . . . 4
119, 10eqeq12d 2109 . . 3
127, 8, 11cbvral 2600 . 2
131, 12syl6bb 195 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1296  wnfc 2222  wral 2370   wfn 5044  cfv 5049 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057 This theorem is referenced by: (None)
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