Theorem predeq1 6314

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

Step	Hyp	Ref	Expression
1		eqid 2726	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2726	. 2 ⊢ 𝑋 = 𝑋
3		predeq123 6313	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
4	1, 2, 3	mp3an23 1450	1 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))

Syntax hints:   → wi 4   = wceq 1534  Predcpred 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312

This theorem is referenced by:  wrecseq123OLD  8330