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Theorem predeq123 29462
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)
Assertion
Ref Expression
predeq123  |-  ( ( R  =  S  /\  A  =  B  /\  X  =  Y )  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  B ,  Y )
)

Proof of Theorem predeq123
StepHypRef Expression
1 simp2 997 . . 3  |-  ( ( R  =  S  /\  A  =  B  /\  X  =  Y )  ->  A  =  B )
2 cnveq 5186 . . . . 5  |-  ( R  =  S  ->  `' R  =  `' S
)
323ad2ant1 1017 . . . 4  |-  ( ( R  =  S  /\  A  =  B  /\  X  =  Y )  ->  `' R  =  `' S )
4 sneq 4042 . . . . 5  |-  ( X  =  Y  ->  { X }  =  { Y } )
543ad2ant3 1019 . . . 4  |-  ( ( R  =  S  /\  A  =  B  /\  X  =  Y )  ->  { X }  =  { Y } )
63, 5imaeq12d 5348 . . 3  |-  ( ( R  =  S  /\  A  =  B  /\  X  =  Y )  ->  ( `' R " { X } )  =  ( `' S " { Y } ) )
71, 6ineq12d 3697 . 2  |-  ( ( R  =  S  /\  A  =  B  /\  X  =  Y )  ->  ( A  i^i  ( `' R " { X } ) )  =  ( B  i^i  ( `' S " { Y } ) ) )
8 df-pred 29461 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
9 df-pred 29461 . 2  |-  Pred ( S ,  B ,  Y )  =  ( B  i^i  ( `' S " { Y } ) )
107, 8, 93eqtr4g 2523 1  |-  ( ( R  =  S  /\  A  =  B  /\  X  =  Y )  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  B ,  Y )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    i^i cin 3470   {csn 4032   `'ccnv 5007   "cima 5011   Predcpred 29460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-pred 29461
This theorem is referenced by:  predeq1  29463  predeq2  29464  predeq3  29465  wsuceq123  29587  wlimeq12  29592
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