Theorem oprabbii 6023

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 2207	. 2 |- w = w
2		oprabbii.1	. . . 4 |- ( ph <-> ps )
3	2	a1i 9	. . 3 |- ( w = w -> ( ph <-> ps ) )
4	3	oprabbidv 6022	. 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 105   = wceq 1373   {coprab 5968

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-oprab 5971

This theorem is referenced by:  oprab4  6039  mpov  6058  dfxp3  6303  tposmpo  6390  oviec  6751  dfplpq2  7502  dfmpq2  7503  dfmq0qs  7577  dfplq0qs  7578  addsrpr  7893  mulsrpr  7894  addcnsr  7982  mulcnsr  7983  addvalex  7992