Theorem bj-rabtrALT 32927

Description: Alternate proof of bj-rabtr 32926. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-rabtrALT	|- { x e. A | T. } = A

Distinct variable group:   x, A

Proof of Theorem bj-rabtrALT

Step	Hyp	Ref	Expression
1		nfrab1 3122	. . 3 |- F/_ x { x e. A | T. }
2		nfcv 2764	. . 3 |- F/_ x A
3	1, 2	cleqf 2790	. 2 |- ( { x e. A | T. } = A <-> A. x ( x e. { x e. A | T. } <-> x e. A ) )
4		tru 1487	. . 3 |- T.
5		rabid 3116	. . 3 |- ( x e. { x e. A | T. } <-> ( x e. A /\ T. ) )
6	4, 5	mpbiran2 954	. 2 |- ( x e. { x e. A | T. } <-> x e. A )
7	3, 6	mpgbir 1726	1 |- { x e. A | T. } = A

Syntax hints:   <-> wb 196   = wceq 1483   T. wtru 1484   e. wcel 1990   {crab 2916

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-rab 2921

This theorem is referenced by: (None)