Theorem oprabbidv 5999

Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.)

Hypothesis

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

Assertion

Ref	Expression
oprabbidv	|- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Distinct variable groups:   x, z, ph   y, z, ph

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

Proof of Theorem oprabbidv

Step	Hyp	Ref	Expression
1		nfv 1551	. 2 |- F/ x ph
2		nfv 1551	. 2 |- F/ y ph
3		nfv 1551	. 2 |- F/ z ph
4		oprabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
5	1, 2, 3, 4	oprabbid 5998	1 |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   {coprab 5945

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-oprab 5948

This theorem is referenced by:  oprabbii  6000  mpoeq123dva  6006  mpoeq3dva  6009  resoprab2  6042  erovlem  6714