Theorem oprabbidv 5972

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 1539	. 2 |- F/ x ph
2		nfv 1539	. 2 |- F/ y ph
3		nfv 1539	. 2 |- F/ z ph
4		oprabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
5	1, 2, 3, 4	oprabbid 5971	1 |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   {coprab 5919

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-oprab 5922

This theorem is referenced by:  oprabbii  5973  mpoeq123dva  5979  mpoeq3dva  5982  resoprab2  6015  erovlem  6681