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Theorem eqfnfvd 5211
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 2386 . 2 (φx A (𝐹x) = (𝐺x))
3 eqfnfvd.1 . . 3 (φ𝐹 Fn A)
4 eqfnfvd.2 . . 3 (φ𝐺 Fn A)
5 eqfnfv 5208 . . 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 1242   wcel 1390  wral 2300   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  foeqcnvco  5373  f1eqcocnv  5374  tfrlem1  5864  frecrdg  5931  iseqfeq2  8906
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