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Metamath Proof Explorer

Theorem sbtrt

Description: Partially closed form of sbtr . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ, 4-Jun-2019) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sbtrt.nf	$⊢ Ⅎ y φ$
	Assertion	sbtrt	$⊢ \forall y [y/ x] φ \to φ$

Proof

Step	Hyp	Ref	Expression
1		sbtrt.nf	$⊢ Ⅎ y φ$
2		stdpc4	$⊢ \forall y [y/ x] φ \to [x/ y] [y/ x] φ$
3	1	sbid2	$⊢ [x/ y] [y/ x] φ \leftrightarrow φ$
4	2 3	sylib	$⊢ \forall y [y/ x] φ \to φ$