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Metamath Proof Explorer

Theorem csbeq1a

Description: Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005)

		Ref	Expression
	Assertion	csbeq1a	⊢ ( 𝑥 = 𝐴 → 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )

Step	Hyp	Ref	Expression
1		csbid	⊢ ⦋ 𝑥 / 𝑥 ⦌ 𝐵 = 𝐵
2		csbeq1	⊢ ( 𝑥 = 𝐴 → ⦋ 𝑥 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
3	1 2	eqtr3id	⊢ ( 𝑥 = 𝐴 → 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )