Theorem abbid 2322

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 1545	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		abbi 2319	. 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 1371   = wceq 1373   F/wnf 1483   {cab 2191

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198

This theorem is referenced by:  abbidv  2323  rabeqf  2762  sbcbid  3056