Theorem iota2 5344

Description: The unique element such that ph. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Hypothesis

Ref	Expression
iota2.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
iota2	|- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem iota2

Step	Hyp	Ref	Expression
1		elex 2827	. 2 |- ( A e. B -> A e. _V )
2		simpl 109	. . 3 |- ( ( A e. _V /\ E! x ph ) -> A e. _V )
3		simpr 110	. . 3 |- ( ( A e. _V /\ E! x ph ) -> E! x ph )
4		iota2.1	. . . 4 |- ( x = A -> ( ph <-> ps ) )
5	4	adantl 277	. . 3 |- ( ( ( A e. _V /\ E! x ph ) /\ x = A ) -> ( ph <-> ps ) )
6		nfv 1577	. . . 4 |- F/ x A e. _V
7		nfeu1 2093	. . . 4 |- F/ x E! x ph
8	6, 7	nfan 1614	. . 3 |- F/ x ( A e. _V /\ E! x ph )
9		nfvd 1578	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/ x ps )
10		nfcvd 2387	. . 3 |- ( ( A e. _V /\ E! x ph ) -> F/_ x A )
11	2, 3, 5, 8, 9, 10	iota2df 5340	. 2 |- ( ( A e. _V /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
12	1, 11	sylan 283	1 |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E!weu 2082   e. wcel 2205   _Vcvv 2815   iotacio 5312

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-sn 3697  df-pr 3698  df-uni 3917  df-iota 5314

This theorem is referenced by:  iotam  5346  fvmbr  5707  pczpre  12999  pcdiv  13004  gsum0g  13626  gsumval2  13627  gfsumval  16879