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Theorem feq123d 5358
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq12d.1 (𝜑𝐹 = 𝐺)
feq12d.2 (𝜑𝐴 = 𝐵)
feq123d.3 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
feq123d (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))

Proof of Theorem feq123d
StepHypRef Expression
1 feq12d.1 . . 3 (𝜑𝐹 = 𝐺)
2 feq12d.2 . . 3 (𝜑𝐴 = 𝐵)
31, 2feq12d 5357 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
4 feq123d.3 . . 3 (𝜑𝐶 = 𝐷)
5 feq3 5352 . . 3 (𝐶 = 𝐷 → (𝐺:𝐵𝐶𝐺:𝐵𝐷))
64, 5syl 14 . 2 (𝜑 → (𝐺:𝐵𝐶𝐺:𝐵𝐷))
73, 6bitrd 188 1 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wf 5214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222
This theorem is referenced by:  feq123  5359  feq23d  5363
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