Theorem abbid 2287

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 1515	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		abbi 2284	. 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 1346   = wceq 1348   F/wnf 1453   {cab 2156

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163

This theorem is referenced by:  abbidv  2288  rabeqf  2720  sbcbid  3012