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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 279 1 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  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  7153  smoeq  8350  oif  9525  1fv  13620  catcisolem  18060  hofcl  18212  dmdprd  19868  dpjf  19927  pjf2  21269  mat1dimmul  21978  lmbr2  22763  lmff  22805  dfac14  23122  lmmbr2  24776  lmcau  24830  perfdvf  25420  dvnfre  25469  dvle  25524  dvfsumle  25538  dvfsumge  25539  dvmptrecl  25541  uhgr0e  28331  uhgrstrrepe  28338  incistruhgr  28339  upgr1e  28373  1hevtxdg1  28763  umgr2v2e  28782  iswlk  28867  0wlkons1  29374  resf1o  31955  ismeas  33197  omsmeas  33322  breprexplema  33642  satfun  34402  gg-dvfsumle  35182  mbfresfi  36534  sdclem1  36611  dfac21  41808  fnlimfvre  44390  climrescn  44464  fourierdlem74  44896  fourierdlem103  44925  fourierdlem104  44926  sge0iunmpt  45134  ismea  45167  isome  45210  sssmf  45454  smflimlem3  45489  smflimlem4  45490  isupwlk  46514
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