Theorem riotabidva 5638

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2608 analog.) (Contributed by NM, 17-Jan-2012.)

Hypothesis

Ref	Expression
riotabidva.1	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
riotabidva	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem riotabidva

Step	Hyp	Ref	Expression
1		riotabidva.1	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	pm5.32da 441	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
3	2	iotabidv 5014	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. A /\ ch ) ) )
4		df-riota 5622	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
5		df-riota 5622	. 2 |- ( iota_ x e. A ch ) = ( iota x ( x e. A /\ ch ) )
6	3, 4, 5	3eqtr4g 2146	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   e. wcel 1439   iotacio 4991   iota_crio 5621

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-uni 3660  df-iota 4993  df-riota 5622

This theorem is referenced by:  riotabiia  5639  divfnzn  9160