Theorem riotaeqdv 5971

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 2301	. . . 4 |- ( ph -> ( x e. A <-> x e. B ) )
3	2	anbi1d 465	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ps ) ) )
4	3	iotabidv 5309	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. B /\ ps ) ) )
5		df-riota 5970	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 5970	. 2 |- ( iota_ x e. B ps ) = ( iota x ( x e. B /\ ps ) )
7	4, 5, 6	3eqtr4g 2289	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   iotacio 5284   iota_crio 5969

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-uni 3894  df-iota 5286  df-riota 5970

This theorem is referenced by:  riotaeqbidv  5973  grpinvpropdg  13657