Theorem rzalf 39176

Description: A version of rzal 4073 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

Step	Hyp	Ref	Expression
1		rzalf.1	. 2 |- F/ x A = (/)
2		ne0i 3921	. . . 4 |- ( x e. A -> A =/= (/) )
3	2	necon2bi 2824	. . 3 |- ( A = (/) -> -. x e. A )
4	3	pm2.21d 118	. 2 |- ( A = (/) -> ( x e. A -> ph ) )
5	1, 4	ralrimi 2957	1 |- ( A = (/) -> A. x e. A ph )

Syntax hints:   -> wi 4   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  stoweidlem18  40235  stoweidlem28  40245  stoweidlem55  40272