Theorem riotaeqdv 5489

Description: Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)

Hypothesis

Ref	Expression
riotaeqdv.1	|- ( ph -> A = B )

Assertion

Ref	Expression
riotaeqdv	|- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) )

Distinct variable group:   ph, x

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

Proof of Theorem riotaeqdv

Step	Hyp	Ref	Expression
1		riotaeqdv.1	. . . . 5 |- ( ph -> A = B )
2	1	eleq2d 2148	. . . 4 |- ( ph -> ( x e. A <-> x e. B ) )
3	2	anbi1d 452	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ps ) ) )
4	3	iotabidv 4908	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. B /\ ps ) ) )
5		df-riota 5488	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 5488	. 2 |- ( iota_ x e. B ps ) = ( iota x ( x e. B /\ ps ) )
7	4, 5, 6	3eqtr4g 2138	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   iotacio 4885   iota_crio 5487

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by:  riotaeqbidv  5491