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Metamath Proof Explorer

Theorem sbab

Description: The right-hand side of the second equality is a way of representing proper substitution of y for x into a class variable. (Contributed by NM, 14-Sep-2003)

		Ref	Expression
	Assertion	sbab	⊢ ( 𝑥 = 𝑦 → 𝐴 = { 𝑧 ∣ [ 𝑦 / 𝑥 ] 𝑧 ∈ 𝐴 } )

Proof

Step	Hyp	Ref	Expression
1		sbequ12	⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝐴 ↔ [ 𝑦 / 𝑥 ] 𝑧 ∈ 𝐴 ) )
2	1	eqabdv	⊢ ( 𝑥 = 𝑦 → 𝐴 = { 𝑧 ∣ [ 𝑦 / 𝑥 ] 𝑧 ∈ 𝐴 } )