Theorem oprabbid 5558

Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
oprabbid.1	|- F/ x ph
oprabbid.2	|- F/ y ph
oprabbid.3	|- F/ z ph
oprabbid.4	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
oprabbid	|- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)   ch( x, y, z)

Proof of Theorem oprabbid

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oprabbid.1	. . . 4 |- F/ x ph
2		oprabbid.2	. . . . 5 |- F/ y ph
3		oprabbid.3	. . . . . 6 |- F/ z ph
4		oprabbid.4	. . . . . . 7 |- ( ph -> ( ps <-> ch ) )
5	4	anbi2d 437	. . . . . 6 |- ( ph -> ( ( w = <. <. x , y >. , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ch ) ) )
6	3, 5	exbid 1507	. . . . 5 |- ( ph -> ( E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
7	2, 6	exbid 1507	. . . 4 |- ( ph -> ( E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
8	1, 7	exbid 1507	. . 3 |- ( ph -> ( E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
9	8	abbidv 2155	. 2 |- ( ph -> { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) } )
10		df-oprab 5516	. 2 |- { <. <. x , y >. , z >. | ps } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) }
11		df-oprab 5516	. 2 |- { <. <. x , y >. , z >. | ch } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) }
12	9, 10, 11	3eqtr4g 2097	1 |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   E.wex 1381   {cab 2026   <.cop 3378   {coprab 5513

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-oprab 5516

This theorem is referenced by:  oprabbidv  5559  mpt2eq123  5564