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Theorem feq12d 6716
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 6713 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐴𝐶))
3 feq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43feq2d 6714 . 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 1534  wf 6550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-fun 6556  df-fn 6557  df-f 6558
This theorem is referenced by:  feq123d  6717  fprg  7169  smoeq  8380  oif  9573  1fv  13674  catcisolem  18132  hofcl  18284  dmdprd  19998  dpjf  20057  pjf2  21712  mat1dimmul  22469  lmbr2  23254  lmff  23296  dfac14  23613  lmmbr2  25278  lmcau  25332  perfdvf  25923  dvnfre  25975  dvle  26031  dvfsumle  26045  dvfsumleOLD  26046  dvfsumge  26047  dvmptrecl  26049  uhgr0e  29007  uhgrstrrepe  29014  incistruhgr  29015  upgr1e  29049  1hevtxdg1  29443  umgr2v2e  29462  iswlk  29547  0wlkons1  30054  resf1o  32644  ismeas  34032  omsmeas  34157  breprexplema  34476  satfun  35239  mbfresfi  37367  sdclem1  37444  dfac21  42727  fnlimfvre  45295  climrescn  45369  fourierdlem74  45801  fourierdlem103  45830  fourierdlem104  45831  sge0iunmpt  46039  ismea  46072  isome  46115  sssmf  46359  smflimlem3  46394  smflimlem4  46395  isupwlk  47513
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