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

Metamath Proof Explorer

Theorem sb4e

Description: One direction of a simplified definition of substitution that unlike sb4b does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

		Ref	Expression
	Assertion	sb4e	$⊢ [y/ x] φ \to \forall x (x = y \to \exists y φ)$

Proof

Step	Hyp	Ref	Expression
1		sb1	$⊢ [y/ x] φ \to \exists x (x = y \land φ)$
2		equs5e	$⊢ \exists x (x = y \land φ) \to \forall x (x = y \to \exists y φ)$
3	1 2	syl	$⊢ [y/ x] φ \to \forall x (x = y \to \exists y φ)$