Theorem bj-abbid 32778

Description: Remove dependency on ax-13 2246 from abbid 2740. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-abbid.1	|- F/ x ph
bj-abbid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
bj-abbid	|- ( ph -> { x | ps } = { x | ch } )

Proof of Theorem bj-abbid

Step	Hyp	Ref	Expression
1		bj-abbid.1	. . 3 |- F/ x ph
2		bj-abbid.2	. . 3 |- ( ph -> ( ps <-> ch ) )
3	1, 2	alrimi 2082	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		bj-abbi 32775	. 2 |- ( A. x ( ps <-> ch ) <-> { x | ps } = { x | ch } )
5	3, 4	sylib 208	1 |- ( ph -> { x | ps } = { x | ch } )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   F/wnf 1708   {cab 2608

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-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

This theorem is referenced by:  bj-abbidv  32779