Theorem trpredeq1d 33057

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 33054	. 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 1533  TrPredctrpred 33051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-xp 5555  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-iota 6308  df-fv 6357  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-trpred 33052

This theorem is referenced by: (None)