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Theorem rgenz 3933
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 3929 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
2 rgenz.1 . . 3  |-  ( ( A  =/=  (/)  /\  x  e.  A )  ->  ph )
32ralrimiva 2878 . 2  |-  ( A  =/=  (/)  ->  A. x  e.  A  ph )
41, 3pm2.61ine 2780 1  |-  A. x  e.  A  ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767    =/= wne 2662   A.wral 2814   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-v 3115  df-dif 3479  df-nul 3786
This theorem is referenced by: (None)
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