This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem hbsb3

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 bj-hbsb3v for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	hbsb3.1	$⊢ φ \to \forall y φ$
	Assertion	hbsb3	$⊢ [y/ x] φ \to \forall x [y/ x] φ$

Proof

Step	Hyp	Ref	Expression
1		hbsb3.1	$⊢ φ \to \forall y φ$
2	1	sbimi	$⊢ [y/ x] φ \to [y/ x] \forall y φ$
3		hbsb2a	$⊢ [y/ x] \forall y φ \to \forall x [y/ x] φ$
4	2 3	syl	$⊢ [y/ x] φ \to \forall x [y/ x] φ$