Theorem abbid 2348

Description: Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem abbid

Step	Hyp	Ref	Expression
1		abbid.1	. . 3 |- F/ x ph
2		abbid.2	. . 3 |- ( ph -> ( ps <-> ch ) )
3	1, 2	alrimi 1570	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		abbi 2345	. 2 |- ( A. x ( ps <-> ch ) <-> { x | ps } = { x | ch } )
5	3, 4	sylib 122	1 |- ( ph -> { x | ps } = { x | ch } )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1395   = wceq 1397   F/wnf 1508   {cab 2217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224

This theorem is referenced by:  abbidv  2349  rabeqf  2792  sbcbid  3089