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Theorem predeq2 29222
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predeq2  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )

Proof of Theorem predeq2
StepHypRef Expression
1 eqid 2443 . 2  |-  R  =  R
2 eqid 2443 . 2  |-  X  =  X
3 predeq123 29220 . 2  |-  ( ( R  =  R  /\  A  =  B  /\  X  =  X )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X )
)
41, 2, 3mp3an13 1316 1  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383   Predcpred 29218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-pred 29219
This theorem is referenced by:  prednn  29256  prednn0  29257  trpredeq2  29279  frmin  29297  wrecseq123  29312  wfrlem5  29322  frrlem5  29366
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