Theorem abbid 2283

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 1510	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		abbi 2280	. 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 1341   = wceq 1343   F/wnf 1448   {cab 2151

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158

This theorem is referenced by:  abbidv  2284  rabeqf  2716  sbcbid  3008