Theorem oprabbii 5897

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 2165	. 2 |- w = w
2		oprabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( w = w -> ( ph <-> ps ) )
4	3	oprabbidv 5896	. 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 1343   {coprab 5843

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-oprab 5846

This theorem is referenced by:  oprab4  5913  mpov  5932  dfxp3  6162  tposmpo  6249  oviec  6607  dfplpq2  7295  dfmpq2  7296  dfmq0qs  7370  dfplq0qs  7371  addsrpr  7686  mulsrpr  7687  addcnsr  7775  mulcnsr  7776  addvalex  7785