Theorem trpredeq3d 33367

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

Hypothesis

Ref	Expression
trpredeq3d.1	⊢ (𝜑 → 𝑋 = 𝑌)

Assertion

Ref	Expression
trpredeq3d	⊢ (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Proof of Theorem trpredeq3d

Step	Hyp	Ref	Expression
1		trpredeq3d.1	. 2 ⊢ (𝜑 → 𝑋 = 𝑌)
2		trpredeq3 33364	. 2 ⊢ (𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))
3	1, 2	syl 17	1 ⊢ (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1542  TrPredctrpred 33359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5531  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-iota 6297  df-fv 6347  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-trpred 33360

This theorem is referenced by: (None)