Theorem iotain 38618

Description: Equivalence between two different forms of iota. (Contributed by Andrew Salmon, 15-Jul-2011.)

Assertion

Ref	Expression
iotain	|- ( E! x ph -> |^| { x | ph } = ( iota x ph ) )

Proof of Theorem iotain

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2474	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		vex 3203	. . . . 5 |- y e. _V
3	2	intsn 4513	. . . 4 |- |^| { y } = y
4		nfa1 2028	. . . . . . 7 |- F/ x A. x ( ph <-> x = y )
5		sp 2053	. . . . . . 7 |- ( A. x ( ph <-> x = y ) -> ( ph <-> x = y ) )
6	4, 5	abbid 2740	. . . . . 6 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { x | x = y } )
7		df-sn 4178	. . . . . 6 |- { y } = { x | x = y }
8	6, 7	syl6eqr 2674	. . . . 5 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { y } )
9	8	inteqd 4480	. . . 4 |- ( A. x ( ph <-> x = y ) -> |^| { x | ph } = |^| { y } )
10		iotaval 5862	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
11	3, 9, 10	3eqtr4a 2682	. . 3 |- ( A. x ( ph <-> x = y ) -> |^| { x | ph } = ( iota x ph ) )
12	11	exlimiv 1858	. 2 |- ( E. y A. x ( ph <-> x = y ) -> |^| { x | ph } = ( iota x ph ) )
13	1, 12	sylbi 207	1 |- ( E! x ph -> |^| { x | ph } = ( iota x ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   E!weu 2470   {cab 2608   {csn 4177   |^|cint 4475   iotacio 5849

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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-in 3581  df-sn 4178  df-pr 4180  df-uni 4437  df-int 4476  df-iota 5851

This theorem is referenced by: (None)