Theorem oprabbidv 6077

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 1576	. 2 |- F/ x ph
2		nfv 1576	. 2 |- F/ y ph
3		nfv 1576	. 2 |- F/ z ph
4		oprabbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
5	1, 2, 3, 4	oprabbid 6076	1 |- ( ph -> { <. <. x , y >. , z >. | ps } = { <. <. x , y >. , z >. | ch } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   {coprab 6021

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-oprab 6024

This theorem is referenced by:  oprabbii  6078  mpoeq123dva  6084  mpoeq3dva  6087  resoprab2  6120  erovlem  6798