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Metamath Proof Explorer

Theorem sbiedv

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie ). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sbiedvw when possible. (Contributed by NM, 7-Jan-2017) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sbiedv.1	$⊢ (φ \land x = y) \to (ψ \leftrightarrow χ)$
	Assertion	sbiedv	$⊢ φ \to ([y/ x] ψ \leftrightarrow χ)$

Proof

Step	Hyp	Ref	Expression
1		sbiedv.1	$⊢ (φ \land x = y) \to (ψ \leftrightarrow χ)$
2		nfv	$⊢ Ⅎ x φ$
3		nfvd	$⊢ φ \to Ⅎ x χ$
4	1	ex	$⊢ φ \to (x = y \to (ψ \leftrightarrow χ))$
5	2 3 4	sbied	$⊢ φ \to ([y/ x] ψ \leftrightarrow χ)$