Theorem trpredeq1d

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

	Ref	Expression
Hypothesis	trpredeq1d.1	⊢ ( 𝜑 → 𝑅 = 𝑆 )
Assertion	trpredeq1d	⊢ ( 𝜑 → TrPred ( 𝑅 , 𝐴 , 𝑋 ) = TrPred ( 𝑆 , 𝐴 , 𝑋 ) )

Proof

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