Theorem oprabbii 5908

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 2170	. 2 |- w = w
2		oprabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( w = w -> ( ph <-> ps ) )
4	3	oprabbidv 5907	. 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 1348   {coprab 5854

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-oprab 5857

This theorem is referenced by:  oprab4  5924  mpov  5943  dfxp3  6173  tposmpo  6260  oviec  6619  dfplpq2  7316  dfmpq2  7317  dfmq0qs  7391  dfplq0qs  7392  addsrpr  7707  mulsrpr  7708  addcnsr  7796  mulcnsr  7797  addvalex  7806