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Theorem spimeh 1679
Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
Hypotheses
Ref Expression
spimeh.1  |-  ( ph  ->  A. x ph )
spimeh.2  |-  ( x  =  z  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spimeh  |-  ( ph  ->  E. x ps )
Distinct variable group:    x, z
Allowed substitution hints:    ph( x, z)    ps( x, z)

Proof of Theorem spimeh
StepHypRef Expression
1 spimeh.1 . 2  |-  ( ph  ->  A. x ph )
2 a9ev 1668 . . . 4  |-  E. x  x  =  z
3 spimeh.2 . . . 4  |-  ( x  =  z  ->  ( ph  ->  ps ) )
42, 3eximii 1587 . . 3  |-  E. x
( ph  ->  ps )
5419.35i 1611 . 2  |-  ( A. x ph  ->  E. x ps )
61, 5syl 16 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   E.wex 1550
This theorem is referenced by:  ax12olem1OLD  2011  ax10lem2OLD  2026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-9 1666
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
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