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Theorem rgenz 3572
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 3568 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
2 rgenz.1 . . 3  |-  ( ( A  =/=  (/)  /\  x  e.  A )  ->  ph )
32ralrimiva 2639 . 2  |-  ( A  =/=  (/)  ->  A. x  e.  A  ph )
41, 3pm2.61ine 2535 1  |-  A. x  e.  A  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    =/= wne 2459   A.wral 2556   (/)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-v 2803  df-dif 3168  df-nul 3469
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