Metamath Proof Explorer


Theorem spime

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

Ref Expression
Hypotheses spime.1
|- F/ x ph
spime.2
|- ( x = y -> ( ph -> ps ) )
Assertion spime
|- ( ph -> E. x ps )

Proof

Step Hyp Ref Expression
1 spime.1
 |-  F/ x ph
2 spime.2
 |-  ( x = y -> ( ph -> ps ) )
3 1 a1i
 |-  ( T. -> F/ x ph )
4 3 2 spimed
 |-  ( T. -> ( ph -> E. x ps ) )
5 4 mptru
 |-  ( ph -> E. x ps )