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Theorem feq12d 5545
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq12d.1  |-  ( ph  ->  F  =  G )
feq12d.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
feq12d  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )

Proof of Theorem feq12d
StepHypRef Expression
1 feq12d.1 . . 3  |-  ( ph  ->  F  =  G )
21feq1d 5543 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : A --> C ) )
3 feq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43feq2d 5544 . 2  |-  ( ph  ->  ( G : A --> C 
<->  G : B --> C ) )
52, 4bitrd 253 1  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1364   -->wf 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-fun 5417  df-fn 5418  df-f 5419
This theorem is referenced by:  feq123d  5546  fprg  5888  smoeq  6807  oif  7740  catcisolem  14970  hofcl  15065  dmdprd  16470  dpjf  16546  pjf2  18039  lmbr2  18763  lmff  18805  dfac14  19091  lmmbr2  20670  lmcau  20723  perfdvf  21278  dvnfre  21326  dvle  21379  dvfsumle  21393  dvfsumge  21394  dvmptrecl  21396  isumgra  23168  iswlk  23345  istrl  23355  constr1trl  23406  constr3trllem1  23455  eupap1  23516  resf1o  25949  ismeas  26533  mbfresfi  28347  sdclem1  28548  dfac21  29328  mat1dimmul  30738
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