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Theorem rexlimib 25062
Description: Removal of a universal restricted quantifier in an antecedent. See also reximdva0 3479. (Contributed by FL, 19-Apr-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
rexlimib.1  |-  F/ x ps
rexlimib.2  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
rexlimib  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rexlimib
StepHypRef Expression
1 r19.2z 3556 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
2 rexlimib.1 . . 3  |-  F/ x ps
3 rexlimib.2 . . 3  |-  ( x  e.  A  ->  ( ph  ->  ps ) )
42, 3rexlimi 2673 . 2  |-  ( E. x  e.  A  ph  ->  ps )
51, 4syl 15 1  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   F/wnf 1534    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   (/)c0 3468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-nul 3469
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