Theorem riotaeqdv 6612

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 2687	. . . 4 |- ( ph -> ( x e. A <-> x e. B ) )
3	2	anbi1d 741	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ps ) ) )
4	3	iotabidv 5872	. 2 |- ( ph -> ( iota x ( x e. A /\ ps ) ) = ( iota x ( x e. B /\ ps ) ) )
5		df-riota 6611	. 2 |- ( iota_ x e. A ps ) = ( iota x ( x e. A /\ ps ) )
6		df-riota 6611	. 2 |- ( iota_ x e. B ps ) = ( iota x ( x e. B /\ ps ) )
7	4, 5, 6	3eqtr4g 2681	1 |- ( ph -> ( iota_ x e. A ps ) = ( iota_ x e. B ps ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   iotacio 5849   iota_crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-uni 4437  df-iota 5851  df-riota 6611

This theorem is referenced by:  riotaeqbidv  6614  grpinvpropd  17490  funtransport  32138  fvtransport  32139