Theorem trpredeq2d

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

	Ref	Expression
Hypothesis	trpredeq2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	trpredeq2d	⊢ ( 𝜑 → TrPred ( 𝑅 , 𝐴 , 𝑋 ) = TrPred ( 𝑅 , 𝐵 , 𝑋 ) )

Proof

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