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Theorem rzalf 27791
Description: A version of rzal 3568 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypothesis
Ref Expression
rzalf.1  |-  F/ x  A  =  (/)
Assertion
Ref Expression
rzalf  |-  ( A  =  (/)  ->  A. x  e.  A  ph )

Proof of Theorem rzalf
StepHypRef Expression
1 rzalf.1 . 2  |-  F/ x  A  =  (/)
2 ne0i 3474 . . . 4  |-  ( x  e.  A  ->  A  =/=  (/) )
32necon2bi 2505 . . 3  |-  ( A  =  (/)  ->  -.  x  e.  A )
43pm2.21d 98 . 2  |-  ( A  =  (/)  ->  ( x  e.  A  ->  ph )
)
51, 4ralrimi 2637 1  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1534    = wceq 1632    e. wcel 1696   A.wral 2556   (/)c0 3468
This theorem is referenced by:  stoweidlem18  27870  stoweidlem28  27880  stoweidlem55  27907
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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