Theorem iota4 4358

Description: Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iota4	|- (E!xph -> [.(iotaxph) / x ].ph)

Proof of Theorem iota4

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		bi2 189	. . . . . 6 |- ((ph <-> x = z) -> (x = z -> ph))
3	2	alimi 1559	. . . . 5 |- (A.x(ph <-> x = z) -> A.x(x = z -> ph))
4		sb2 2023	. . . . 5 |- (A.x(x = z -> ph) -> [z / x]ph)
5	3, 4	syl 15	. . . 4 |- (A.x(ph <-> x = z) -> [z / x]ph)
6		iotaval 4351	. . . . . 6 |- (A.x(ph <-> x = z) -> (iotaxph) = z)
7	6	eqcomd 2358	. . . . 5 |- (A.x(ph <-> x = z) -> z = (iotaxph))
8		dfsbcq2 3050	. . . . 5 |- (z = (iotaxph) -> ([z / x]ph <-> [.(iotaxph) / x ].ph))
9	7, 8	syl 15	. . . 4 |- (A.x(ph <-> x = z) -> ([z / x]ph <-> [.(iotaxph) / x ].ph))
10	5, 9	mpbid 201	. . 3 |- (A.x(ph <-> x = z) -> [.(iotaxph) / x ].ph)
11	10	exlimiv 1634	. 2 |- (E.zA.x(ph <-> x = z) -> [.(iotaxph) / x ].ph)
12	1, 11	sylbi 187	1 |- (E!xph -> [.(iotaxph) / x ].ph)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642  [wsb 1648  E!weu 2204   [.wsbc 3047  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:  iota4an  4359  iotacl  4363