Theorem iota2 4368

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!xph) -> (ps <-> (iotaxph) = 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 2868	. 2 |- (A e. B -> A e. _V)
2		simpl 443	. . 3 |- ((A e. _V /\ E!xph) -> A e. _V)
3		simpr 447	. . 3 |- ((A e. _V /\ E!xph) -> E!xph)
4		iota2.1	. . . 4 |- (x = A -> (ph <-> ps))
5	4	adantl 452	. . 3 |- (((A e. _V /\ E!xph) /\ x = A) -> (ph <-> ps))
6		nfv 1619	. . . 4 |- F/x A e. _V
7		nfeu1 2214	. . . 4 |- F/xE!xph
8	6, 7	nfan 1824	. . 3 |- F/x(A e. _V /\ E!xph)
9		nfvd 1620	. . 3 |- ((A e. _V /\ E!xph) -> F/xps)
10		nfcvd 2491	. . 3 |- ((A e. _V /\ E!xph) -> F/_xA)
11	2, 3, 5, 8, 9, 10	iota2df 4366	. 2 |- ((A e. _V /\ E!xph) -> (ps <-> (iotaxph) = A))
12	1, 11	sylan 457	1 |- ((A e. B /\ E!xph) -> (ps <-> (iotaxph) = A))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E!weu 2204  _Vcvv 2860  iotacio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by:  reiota2  4369  tz6.12-1  5345