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Theorem feq12d 6705
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 6702 . 2 (𝜑 → (𝐹:𝐴𝐶𝐺:𝐴𝐶))
3 feq12d.2 . . 3 (𝜑𝐴 = 𝐵)
43feq2d 6703 . 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 6539
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 2703
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 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  feq123d  6706  fprg  7152  smoeq  8349  oif  9524  1fv  13619  catcisolem  18059  hofcl  18211  dmdprd  19867  dpjf  19926  pjf2  21268  mat1dimmul  21977  lmbr2  22762  lmff  22804  dfac14  23121  lmmbr2  24775  lmcau  24829  perfdvf  25419  dvnfre  25468  dvle  25523  dvfsumle  25537  dvfsumge  25538  dvmptrecl  25540  uhgr0e  28328  uhgrstrrepe  28335  incistruhgr  28336  upgr1e  28370  1hevtxdg1  28760  umgr2v2e  28779  iswlk  28864  0wlkons1  29371  resf1o  31950  ismeas  33192  omsmeas  33317  breprexplema  33637  satfun  34397  gg-dvfsumle  35177  mbfresfi  36529  sdclem1  36606  dfac21  41798  fnlimfvre  44380  climrescn  44454  fourierdlem74  44886  fourierdlem103  44915  fourierdlem104  44916  sge0iunmpt  45124  ismea  45157  isome  45200  sssmf  45444  smflimlem3  45479  smflimlem4  45480  isupwlk  46504
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