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

Metamath Proof Explorer

Theorem sbid2

Description: An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out sbid2vw for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sbid2.1	$⊢ Ⅎ x φ$
	Assertion	sbid2	$⊢ [y/ x] [x/ y] φ \leftrightarrow φ$

Proof

Step	Hyp	Ref	Expression
1		sbid2.1	$⊢ Ⅎ x φ$
2		sbco	$⊢ [y/ x] [x/ y] φ \leftrightarrow [y/ x] φ$
3	1	sbf	$⊢ [y/ x] φ \leftrightarrow φ$
4	2 3	bitri	$⊢ [y/ x] [x/ y] φ \leftrightarrow φ$