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Metamath Proof Explorer

Theorem sbcex

Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016)

		Ref	Expression
	Assertion	sbcex	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		df-sbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜑 } )
2		elex	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } → 𝐴 ∈ V )
3	1 2	sylbi	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → 𝐴 ∈ V )