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Theorem predeq2 25442
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 2436 . 2  |-  R  =  R
2 eqid 2436 . 2  |-  X  =  X
3 predeq123 25440 . 2  |-  ( ( R  =  R  /\  A  =  B  /\  X  =  X )  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X )
)
41, 2, 3mp3an13 1270 1  |-  ( A  =  B  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  B ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   Predcpred 25438
This theorem is referenced by:  prednn  25476  prednn0  25477  trpredeq2  25499  frmin  25517  wrecseq123  25532  wfrlem5  25542  frrlem5  25586
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-pred 25439
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