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Theorem r19.37v 2951
Description: Restricted quantifier version of one direction of 19.37v 1836. (The other direction holds iff  A is nonempty, see r19.37zv 3876.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.37v  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( ph  ->  E. x  e.  A  ps )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem r19.37v
StepHypRef Expression
1 nfv 1771 . 2  |-  F/ x ph
21r19.37 2950 1  |-  ( E. x  e.  A  (
ph  ->  ps )  -> 
( ph  ->  E. x  e.  A  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wrex 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-ral 2753  df-rex 2754
This theorem is referenced by:  ssiun  4333  isucn2  21342
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