Theorem predeq2 6300

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

Step	Hyp	Ref	Expression
1		eqid 2732	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2732	. 2 ⊢ 𝑋 = 𝑋
3		predeq123 6298	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
4	1, 2, 3	mp3an13 1452	1 ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   = wceq 1541  Predcpred 6296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297

This theorem is referenced by:  fprlem1  8281  wrecseq123OLD  8296  wfrlem5OLD  8309  frmin  9740  frrlem15  9748  prednn  13620  prednn0  13621