Theorem trpredeq1d 32616

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

Hypothesis

Ref	Expression
trpredeq1d.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
trpredeq1d	⊢ (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Proof of Theorem trpredeq1d

Step	Hyp	Ref	Expression
1		trpredeq1d.1	. 2 ⊢ (𝜑 → 𝑅 = 𝑆)
2		trpredeq1 32613	. 2 ⊢ (𝑅 = 𝑆 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))
3	1, 2	syl 17	1 ⊢ (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Syntax hints:   → wi 4   = wceq 1508  TrPredctrpred 32610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-xp 5410  df-cnv 5412  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-iota 6150  df-fv 6194  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-trpred 32611

This theorem is referenced by: (None)