Theorem trpredeq2d 33067

Description: Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Hypothesis

Ref	Expression
trpredeq2d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
trpredeq2d	⊢ (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Proof of Theorem trpredeq2d

Step	Hyp	Ref	Expression
1		trpredeq2d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		trpredeq2 33064	. 2 ⊢ (𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))
3	1, 2	syl 17	1 ⊢ (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   = wceq 1536  TrPredctrpred 33060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-xp 5564  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-iota 6317  df-fv 6366  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-trpred 33061

This theorem is referenced by: (None)