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Theorem eqfnfvd 5214
Description: Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
eqfnfvd.1 (φ𝐹 Fn A)
eqfnfvd.2 (φ𝐺 Fn A)
eqfnfvd.3 ((φ x A) → (𝐹x) = (𝐺x))
Assertion
Ref Expression
eqfnfvd (φ𝐹 = 𝐺)
Distinct variable groups:   x,A   x,𝐹   x,𝐺   φ,x

Proof of Theorem eqfnfvd
StepHypRef Expression
1 eqfnfvd.3 . . 3 ((φ x A) → (𝐹x) = (𝐺x))
21ralrimiva 2389 . 2 (φx A (𝐹x) = (𝐺x))
3 eqfnfvd.1 . . 3 (φ𝐹 Fn A)
4 eqfnfvd.2 . . 3 (φ𝐺 Fn A)
5 eqfnfv 5211 . . 3 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
63, 4, 5syl2anc 391 . 2 (φ → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
72, 6mpbird 156 1 (φ𝐹 = 𝐺)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1243   wcel 1393  wral 2303   Fn wfn 4843  cfv 4848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-br 3759  df-opab 3813  df-mpt 3814  df-id 4024  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-iota 4813  df-fun 4850  df-fn 4851  df-fv 4856
This theorem is referenced by:  foeqcnvco  5376  f1eqcocnv  5377  tfrlem1  5867  frecrdg  5934  iseqfeq2  8997
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