Theorem predeq1

Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011)

	Ref	Expression
Assertion	predeq1	⊢ ( 𝑅 = 𝑆 → Pred ( 𝑅 , 𝐴 , 𝑋 ) = Pred ( 𝑆 , 𝐴 , 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2		eqid	⊢ 𝑋 = 𝑋
3		predeq123	⊢ ( ( 𝑅 = 𝑆 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑋 ) → Pred ( 𝑅 , 𝐴 , 𝑋 ) = Pred ( 𝑆 , 𝐴 , 𝑋 ) )
4	1 2 3	mp3an23	⊢ ( 𝑅 = 𝑆 → Pred ( 𝑅 , 𝐴 , 𝑋 ) = Pred ( 𝑆 , 𝐴 , 𝑋 ) )