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

Metamath Proof Explorer

Theorem spime

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of Tarski p. 70. See spimew and spimevw for weaker versions requiring fewer axioms. (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 6-Mar-2018) Usage of this theorem is discouraged because it depends on ax-13 . Use spimefv instead. (New usage is discouraged.)

	Ref	Expression
Hypotheses	spime.1	$⊢ Ⅎ x φ$
	spime.2	$⊢ x = y \to (φ \to ψ)$
Assertion	spime	$⊢ φ \to \exists x ψ$

Proof

Step	Hyp	Ref	Expression
1		spime.1	$⊢ Ⅎ x φ$
2		spime.2	$⊢ x = y \to (φ \to ψ)$
3	1	a1i	$⊢ ⊤ \to Ⅎ x φ$
4	3 2	spimed	$⊢ ⊤ \to (φ \to \exists x ψ)$
5	4	mptru	$⊢ φ \to \exists x ψ$