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Theorem spimeh 1674
Description: Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
spimeh.1  |-  ( ph  ->  A. x ph )
spimeh.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spimeh  |-  ( ph  ->  E. x ps )

Proof of Theorem spimeh
StepHypRef Expression
1 a9e 1631 . 2  |-  E. x  x  =  y
2 spimeh.1 . . 3  |-  ( ph  ->  A. x ph )
3 spimeh.2 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
43com12 30 . . 3  |-  ( ph  ->  ( x  =  y  ->  ps ) )
52, 4eximdh 1547 . 2  |-  ( ph  ->  ( E. x  x  =  y  ->  E. x ps ) )
61, 5mpi 15 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287    = wceq 1289   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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