Theorem riotaeqdv 5799

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 2236	. . . 4 |- ( ph -> ( x e. A <-> x e. B ) )
3	2	anbi1d 461	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ps ) ) )
4	3	iotabidv 5174	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. B /\ ps ) ) )
5		df-riota 5798	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 5798	. 2 |- ( iota_ x e. B ps ) = ( iota x ( x e. B /\ ps ) )
7	4, 5, 6	3eqtr4g 2224	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   iotacio 5151   iota_crio 5797

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-uni 3790  df-iota 5153  df-riota 5798

This theorem is referenced by:  riotaeqbidv  5801