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Theorem r19.2zr 24957
Description: Quantifying a hypothesis with a universal restricted quantifier. (Contributed by FL, 19-Sep-2011.)
Hypothesis
Ref Expression
r19.2zr.1  |-  ( ph  ->  ps )
Assertion
Ref Expression
r19.2zr  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem r19.2zr
StepHypRef Expression
1 r19.2z 3543 . 2  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
2 r19.2zr.1 . . 3  |-  ( ph  ->  ps )
32rexlimivw 2663 . 2  |-  ( E. x  e.  A  ph  ->  ps )
41, 3syl 15 1  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455
This theorem is referenced by:  r19.2zrr  24958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456
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