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Theorem rgenz 3655
 Description: Generalization rule that eliminates a nonzero class requirement. (Contributed by NM, 8-Dec-2012.)
Hypothesis
Ref Expression
rgenz.1 ((A x A) → φ)
Assertion
Ref Expression
rgenz x A φ
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem rgenz
StepHypRef Expression
1 rzal 3651 . 2 (A = x A φ)
2 rgenz.1 . . 3 ((A x A) → φ)
32ralrimiva 2697 . 2 (Ax A φ)
41, 3pm2.61ine 2592 1 x A φ
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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