Theorem oprabbii 5819

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 2137	. 2 |- w = w
2		oprabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( w = w -> ( ph <-> ps ) )
4	3	oprabbidv 5818	. 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 1331   {coprab 5768

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-oprab 5771

This theorem is referenced by:  oprab4  5835  mpov  5854  dfxp3  6085  tposmpo  6171  oviec  6528  dfplpq2  7155  dfmpq2  7156  dfmq0qs  7230  dfplq0qs  7231  addsrpr  7546  mulsrpr  7547  addcnsr  7635  mulcnsr  7636  addvalex  7645