Theorem oprabbid 5953

Description: Equivalent wff's yield equal operation class abstractions (deduction form). (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 464	. . . . . 6 |- ( ph -> ( ( w = <. <. x , y >. , z >. /\ ps ) <-> ( w = <. <. x , y >. , z >. /\ ch ) ) )
6	3, 5	exbid 1627	. . . . 5 |- ( ph -> ( E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
7	2, 6	exbid 1627	. . . 4 |- ( ph -> ( E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) <-> E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) ) )
8	1, 7	exbid 1627	. . 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 2307	. 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 5904	. 2 |- { <. <. x , y >. , z >. | ps } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ps ) }
11		df-oprab 5904	. 2 |- { <. <. x , y >. , z >. | ch } = { w | E. x E. y E. z ( w = <. <. x , y >. , z >. /\ ch ) }
12	9, 10, 11	3eqtr4g 2247	1 |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1471   E.wex 1503   {cab 2175   <.cop 3613   {coprab 5901

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-oprab 5904

This theorem is referenced by:  oprabbidv  5954  mpoeq123  5959