Theorem riotabidva 5999

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2791 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 452	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
3	2	iotabidv 5316	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. A /\ ch ) ) )
4		df-riota 5981	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
5		df-riota 5981	. 2 |- ( iota_ x e. A ch ) = ( iota x ( x e. A /\ ch ) )
6	3, 4, 5	3eqtr4g 2289	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   iotacio 5291   iota_crio 5980

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-uni 3899  df-iota 5293  df-riota 5981

This theorem is referenced by:  riotabiia  6000  divfnzn  9916  grpinvpropdg  13738  oppr1g  14176