Theorem iota1 4354

Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
iota1	|- (E!xph -> (ph <-> (iotaxph) = x))

Proof of Theorem iota1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2208	. 2 |- (E!xph <-> E.zA.x(ph <-> x = z))
2		sp 1747	. . . . 5 |- (A.x(ph <-> x = z) -> (ph <-> x = z))
3		iotaval 4351	. . . . . 6 |- (A.x(ph <-> x = z) -> (iotaxph) = z)
4	3	eqeq2d 2364	. . . . 5 |- (A.x(ph <-> x = z) -> (x = (iotaxph) <-> x = z))
5	2, 4	bitr4d 247	. . . 4 |- (A.x(ph <-> x = z) -> (ph <-> x = (iotaxph)))
6		eqcom 2355	. . . 4 |- (x = (iotaxph) <-> (iotaxph) = x)
7	5, 6	syl6bb 252	. . 3 |- (A.x(ph <-> x = z) -> (ph <-> (iotaxph) = x))
8	7	exlimiv 1634	. 2 |- (E.zA.x(ph <-> x = z) -> (ph <-> (iotaxph) = x))
9	1, 8	sylbi 187	1 |- (E!xph -> (ph <-> (iotaxph) = x))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642  E!weu 2204  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-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-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:  iota2df  4366  sniota  4370