Theorem abbid 2257

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 1503	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		abbi 2254	. 2 |- ( A. x ( ps <-> ch ) <-> { x | ps } = { x | ch } )
5	3, 4	sylib 121	1 |- ( ph -> { x | ps } = { x | ch } )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1330   = wceq 1332   F/wnf 1437   {cab 2126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133

This theorem is referenced by:  abbidv  2258  rabeqf  2679  sbcbid  2970