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Theorem predeq1 25443
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predeq1  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )

Proof of Theorem predeq1
StepHypRef Expression
1 eqid 2438 . 2  |-  A  =  A
2 eqid 2438 . 2  |-  X  =  X
3 predeq123 25442 . 2  |-  ( ( R  =  S  /\  A  =  A  /\  X  =  X )  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X )
)
41, 2, 3mp3an23 1272 1  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   Predcpred 25440
This theorem is referenced by:  trpredeq1  25500  wrecseq123  25534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 25441
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