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Theorem rgenz 3676
Description: Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.)
Hypothesis
Ref Expression
rgenz.1  |-  ( ( A  =/=  (/)  /\  x  e.  A )  ->  ph )
Assertion
Ref Expression
rgenz  |-  A. x  e.  A  ph
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rgenz
StepHypRef Expression
1 rzal 3672 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
2 rgenz.1 . . 3  |-  ( ( A  =/=  (/)  /\  x  e.  A )  ->  ph )
32ralrimiva 2732 . 2  |-  ( A  =/=  (/)  ->  A. x  e.  A  ph )
41, 3pm2.61ine 2626 1  |-  A. x  e.  A  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717    =/= wne 2550   A.wral 2649   (/)c0 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-v 2901  df-dif 3266  df-nul 3572
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