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Theorem feq12d 6706
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq12d.1 (𝜑𝐹 = 𝐺)
feq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
feq12d (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))

Proof of Theorem feq12d
StepHypRef Expression
1 feq12d.1 . . 3 (𝜑𝐹 = 𝐺)
21feq1d 6703 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐴𝐶))
3 feq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43feq2d 6704 . 2 (𝜑 → (𝐺:𝐴𝐶𝐺:𝐵𝐶))
52, 4bitrd 278 1 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  feq123d  6707  fprg  7156  smoeq  8354  oif  9529  1fv  13626  catcisolem  18066  hofcl  18218  dmdprd  19911  dpjf  19970  pjf2  21490  mat1dimmul  22200  lmbr2  22985  lmff  23027  dfac14  23344  lmmbr2  25009  lmcau  25063  perfdvf  25654  dvnfre  25703  dvle  25758  dvfsumle  25772  dvfsumge  25773  dvmptrecl  25775  uhgr0e  28596  uhgrstrrepe  28603  incistruhgr  28604  upgr1e  28638  1hevtxdg1  29028  umgr2v2e  29047  iswlk  29132  0wlkons1  29639  resf1o  32220  ismeas  33493  omsmeas  33618  breprexplema  33938  satfun  34698  gg-dvfsumle  35470  mbfresfi  36839  sdclem1  36916  dfac21  42112  fnlimfvre  44690  climrescn  44764  fourierdlem74  45196  fourierdlem103  45225  fourierdlem104  45226  sge0iunmpt  45434  ismea  45467  isome  45510  sssmf  45754  smflimlem3  45789  smflimlem4  45790  isupwlk  46814
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