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

Metamath Proof Explorer

Theorem equs5e

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. This proof uses ax12 , see equs5eALT for an alternative one using ax-12 but not ax13 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (Proof shortened by Wolf Lammen, 15-Jan-2018) (New usage is discouraged.)

		Ref	Expression
	Assertion	equs5e	$⊢ \exists x (x = y \land φ) \to \forall x (x = y \to \exists y φ)$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$⊢ Ⅎ x \forall x (x = y \to \exists y φ)$
2		ax12	$⊢ x = y \to (\forall y \exists y φ \to \forall x (x = y \to \exists y φ))$
3		hbe1	$⊢ \exists y φ \to \forall y \exists y φ$
4	3	19.23bi	$⊢ φ \to \forall y \exists y φ$
5	2 4	impel	$⊢ (x = y \land φ) \to \forall x (x = y \to \exists y φ)$
6	1 5	exlimi	$⊢ \exists x (x = y \land φ) \to \forall x (x = y \to \exists y φ)$