Theorem oprabbii 5834

Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypothesis

Ref	Expression
oprabbii.1	|- ( ph <-> ps )

Assertion

Ref	Expression
oprabbii	|- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , z >. | ps }

Distinct variable groups:   x, z   y, z

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

Proof of Theorem oprabbii

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2140	. 2 |- w = w
2		oprabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( w = w -> ( ph <-> ps ) )
4	3	oprabbidv 5833	. 2 |- ( w = w -> { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , z >. | ps } )
5	1, 4	ax-mp 5	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , z >. | ps }

Syntax hints:   <-> wb 104   = wceq 1332   {coprab 5783

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-oprab 5786

This theorem is referenced by:  oprab4  5850  mpov  5869  dfxp3  6100  tposmpo  6186  oviec  6543  dfplpq2  7186  dfmpq2  7187  dfmq0qs  7261  dfplq0qs  7262  addsrpr  7577  mulsrpr  7578  addcnsr  7666  mulcnsr  7667  addvalex  7676