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Metamath Proof Explorer

Theorem nfs1

Description: If y is not free in ph , x is not free in [ y / x ] ph . Usage of this theorem is discouraged because it depends on ax-13 . Check out nfs1v for a version requiring fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	nfs1.1	$⊢ Ⅎ y φ$
	Assertion	nfs1	$⊢ Ⅎ x [y/ x] φ$

Proof

Step	Hyp	Ref	Expression
1		nfs1.1	$⊢ Ⅎ y φ$
2	1	nf5ri	$⊢ φ \to \forall y φ$
3	2	hbsb3	$⊢ [y/ x] φ \to \forall x [y/ x] φ$
4	3	nf5i	$⊢ Ⅎ x [y/ x] φ$