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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 
|- 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 )