Theorem riotaeqdv 5731

Description: Formula-building deduction 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 2209	. . . 4 |- ( ph -> ( x e. A <-> x e. B ) )
3	2	anbi1d 460	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ps ) ) )
4	3	iotabidv 5109	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. B /\ ps ) ) )
5		df-riota 5730	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 5730	. 2 |- ( iota_ x e. B ps ) = ( iota x ( x e. B /\ ps ) )
7	4, 5, 6	3eqtr4g 2197	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   iotacio 5086   iota_crio 5729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-uni 3737  df-iota 5088  df-riota 5730

This theorem is referenced by:  riotaeqbidv  5733