Theorem oprabbidv 6709

Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (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 1843	. 2 |- F/ x ph
2		nfv 1843	. 2 |- F/ y ph
3		nfv 1843	. 2 |- F/ z ph
4		oprabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
5	1, 2, 3, 4	oprabbid 6708	1 |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-oprab 6654

This theorem is referenced by:  oprabbii  6710  mpt2eq123dva  6716  mpt2eq3dva  6719  resoprab2  6757  erovlem  7843  joinfval  17001  meetfval  17015  odumeet  17140  odujoin  17142  mppsval  31469  csbmpt22g  33177  unceq  33386  uncf  33388  unccur  33392