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Theorem rzalf 27349
Description: A version of rzal 3665 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 3570 . . . 4  |-  ( x  e.  A  ->  A  =/=  (/) )
32necon2bi 2589 . . 3  |-  ( A  =  (/)  ->  -.  x  e.  A )
43pm2.21d 100 . 2  |-  ( A  =  (/)  ->  ( x  e.  A  ->  ph )
)
51, 4ralrimi 2723 1  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1550    = wceq 1649    e. wcel 1717   A.wral 2642   (/)c0 3564
This theorem is referenced by:  stoweidlem18  27428  stoweidlem28  27438  stoweidlem55  27465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-v 2894  df-dif 3259  df-nul 3565
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