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Theorem feq12d 6495
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 6492 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐴𝐶))
3 feq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43feq2d 6493 . 2 (𝜑 → (𝐺:𝐴𝐶𝐺:𝐵𝐶))
52, 4bitrd 280 1 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wf 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-fun 6350  df-fn 6351  df-f 6352
This theorem is referenced by:  feq123d  6496  fprg  6909  smoeq  7976  oif  8982  1fv  13014  catcisolem  17354  hofcl  17497  dmdprd  19049  dpjf  19108  pjf2  20786  mat1dimmul  21013  lmbr2  21795  lmff  21837  dfac14  22154  lmmbr2  23789  lmcau  23843  perfdvf  24428  dvnfre  24476  dvle  24531  dvfsumle  24545  dvfsumge  24546  dvmptrecl  24548  uhgr0e  26783  uhgrstrrepe  26790  incistruhgr  26791  upgr1e  26825  1hevtxdg1  27215  umgr2v2e  27234  iswlk  27319  0wlkons1  27827  resf1o  30392  ismeas  31357  omsmeas  31480  breprexplema  31800  satfun  32555  mbfresfi  34819  sdclem1  34899  dfac21  39544  fnlimfvre  41831  climrescn  41905  fourierdlem74  42342  fourierdlem103  42371  fourierdlem104  42372  sge0iunmpt  42577  ismea  42610  isome  42653  sssmf  42892  smflimlem3  42926  smflimlem4  42927  isupwlk  43888
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