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Metamath Proof Explorer

Theorem sbcth2

Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008) (Proof shortened by Mario Carneiro, 13-Oct-2016)

		Ref	Expression
	Hypothesis	sbcth2.1	$⊢ x \in B \to φ$
	Assertion	sbcth2	$⊢ A \in B \to \underset{˙}{[} A / x \underset{˙}{]} φ$

Step	Hyp	Ref	Expression
1		sbcth2.1	$⊢ x \in B \to φ$
2	1	rgen	$⊢ \forall x \in B φ$
3		rspsbc	$⊢ A \in B \to (\forall x \in B φ \to \underset{˙}{[} A / x \underset{˙}{]} φ)$
4	2 3	mpi	$⊢ A \in B \to \underset{˙}{[} A / x \underset{˙}{]} φ$