Theorem predeq1 6250

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 2731	. 2 ⊢ 𝐴 = 𝐴
2		eqid 2731	. 2 ⊢ 𝑋 = 𝑋
3		predeq123 6249	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
4	1, 2, 3	mp3an23 1455	1 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))

Syntax hints:   → wi 4   = wceq 1541  Predcpred 6247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248

This theorem is referenced by: (None)