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Theorem feq12d 6653
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 6650 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐴𝐶))
3 feq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43feq2d 6651 . 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 1541  wf 6489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6495  df-fn 6496  df-f 6497
This theorem is referenced by:  feq123d  6654  fprg  7097  smoeq  8292  oif  9462  1fv  13552  catcisolem  17988  hofcl  18140  dmdprd  19768  dpjf  19827  pjf2  21105  mat1dimmul  21809  lmbr2  22594  lmff  22636  dfac14  22953  lmmbr2  24607  lmcau  24661  perfdvf  25251  dvnfre  25300  dvle  25355  dvfsumle  25369  dvfsumge  25370  dvmptrecl  25372  uhgr0e  27908  uhgrstrrepe  27915  incistruhgr  27916  upgr1e  27950  1hevtxdg1  28340  umgr2v2e  28359  iswlk  28444  0wlkons1  28951  resf1o  31530  ismeas  32667  omsmeas  32792  breprexplema  33112  satfun  33874  mbfresfi  36091  sdclem1  36169  dfac21  41331  fnlimfvre  43847  climrescn  43921  fourierdlem74  44353  fourierdlem103  44382  fourierdlem104  44383  sge0iunmpt  44591  ismea  44624  isome  44667  sssmf  44911  smflimlem3  44946  smflimlem4  44947  isupwlk  45970
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