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Theorem rgenz 3923
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 3919 . 2  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
2 rgenz.1 . . 3  |-  ( ( A  =/=  (/)  /\  x  e.  A )  ->  ph )
32ralrimiva 2868 . 2  |-  ( A  =/=  (/)  ->  A. x  e.  A  ph )
41, 3pm2.61ine 2767 1  |-  A. x  e.  A  ph
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823    =/= wne 2649   A.wral 2804   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-v 3108  df-dif 3464  df-nul 3784
This theorem is referenced by: (None)
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