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Metamath Proof Explorer

Theorem sb6rfv

Description: Reversed substitution. Version of sb6rf requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993) (Revised by Wolf Lammen, 7-Feb-2023)

		Ref	Expression
	Hypothesis	sb6rfv.nf	$⊢ Ⅎ y φ$
	Assertion	sb6rfv	$⊢ φ \leftrightarrow \forall y (y = x \to [y/ x] φ)$

Proof

Step	Hyp	Ref	Expression
1		sb6rfv.nf	$⊢ Ⅎ y φ$
2		sbequ12r	$⊢ y = x \to ([y/ x] φ \leftrightarrow φ)$
3	1 2	equsalv	$⊢ \forall y (y = x \to [y/ x] φ) \leftrightarrow φ$
4	3	bicomi	$⊢ φ \leftrightarrow \forall y (y = x \to [y/ x] φ)$