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Theorem f1eq123d 5661
Description: Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
f1eq123d.1  |-  ( ph  ->  F  =  G )
f1eq123d.2  |-  ( ph  ->  A  =  B )
f1eq123d.3  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
f1eq123d  |-  ( ph  ->  ( F : A -1-1-> C  <-> 
G : B -1-1-> D
) )

Proof of Theorem f1eq123d
StepHypRef Expression
1 f1eq123d.1 . . 3  |-  ( ph  ->  F  =  G )
2 f1eq1 5626 . . 3  |-  ( F  =  G  ->  ( F : A -1-1-> C  <->  G : A -1-1-> C ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( F : A -1-1-> C  <-> 
G : A -1-1-> C
) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 f1eq2 5627 . . 3  |-  ( A  =  B  ->  ( G : A -1-1-> C  <->  G : B -1-1-> C ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( G : A -1-1-> C  <-> 
G : B -1-1-> C
) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 f1eq3 5628 . . 3  |-  ( C  =  D  ->  ( G : B -1-1-> C  <->  G : B -1-1-> D ) )
97, 8syl 16 . 2  |-  ( ph  ->  ( G : B -1-1-> C  <-> 
G : B -1-1-> D
) )
103, 6, 93bitrd 271 1  |-  ( ph  ->  ( F : A -1-1-> C  <-> 
G : B -1-1-> D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   -1-1->wf1 5443
This theorem is referenced by:  fthf1  14106  cofth  14124  usgraeq12d  21377  usgra0v  21383  usgra1v  21401  usgrares1  21416  usgra2pthspth  28248  2spontn0vne  28297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451
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