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Metamath Proof Explorer

Theorem sbcthdv

Description: Deduction version of sbcth . (Contributed by NM, 30-Nov-2005) (Proof shortened by Andrew Salmon, 8-Jun-2011)

		Ref	Expression
	Hypothesis	sbcthdv.1	$⊢ φ \to ψ$
	Assertion	sbcthdv	$⊢ (φ \land A \in V) \to \underset{˙}{[} A / x \underset{˙}{]} ψ$

Step	Hyp	Ref	Expression
1		sbcthdv.1	$⊢ φ \to ψ$
2	1	alrimiv	$⊢ φ \to \forall x ψ$
3		spsbc	$⊢ A \in V \to (\forall x ψ \to \underset{˙}{[} A / x \underset{˙}{]} ψ)$
4	2 3	mpan9	$⊢ (φ \land A \in V) \to \underset{˙}{[} A / x \underset{˙}{]} ψ$