Theorem trpredeq3d

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

	Ref	Expression
Hypothesis	trpredeq3d.1	⊢ ( 𝜑 → 𝑋 = 𝑌 )
Assertion	trpredeq3d	⊢ ( 𝜑 → TrPred ( 𝑅 , 𝐴 , 𝑋 ) = TrPred ( 𝑅 , 𝐴 , 𝑌 ) )

Proof

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