Theorem rabeq0OLD 3960

Description: Obsolete proof of rabeq0 3957 as of 16-Jul-2021. (Contributed by Jeff Madsen, 7-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rabeq0OLD	|- ( { x e. A | ph } = (/) <-> A. x e. A -. ph )

Proof of Theorem rabeq0OLD

Step	Hyp	Ref	Expression
1		ralnex 2992	. 2 |- ( A. x e. A -. ph <-> -. E. x e. A ph )
2		rabn0OLD 3959	. . 3 |- ( { x e. A | ph } =/= (/) <-> E. x e. A ph )
3	2	necon1bbii 2843	. 2 |- ( -. E. x e. A ph <-> { x e. A | ph } = (/) )
4	1, 3	bitr2i 265	1 |- ( { x e. A | ph } = (/) <-> A. x e. A -. ph )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   A.wral 2912   E.wrex 2913   {crab 2916   (/)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-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)