Theorem euunianOLD 16396

Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Moved to iota4an 5101 in main set.mm and may be deleted by mathbox owner. --NM 7-Sep-2011. Note that iota4an 5101 also has shorter proof.)

Assertion

Ref	Expression
euunianOLD	|- (E!x(ph /\ ps) -> [(iotax(ph /\ ps)) / x]ph)

Proof of Theorem euunianOLD

Step	Hyp	Ref	Expression
1		df-eu 1775	. 2 |- (E!x(ph /\ ps) <-> E.yA.x((ph /\ ps) <-> x = y))
2		bi2 166	. . . . . 6 |- (((ph /\ ps) <-> x = y) -> (x = y -> (ph /\ ps)))
3	2	alimi 1338	. . . . 5 |- (A.x((ph /\ ps) <-> x = y) -> A.x(x = y -> (ph /\ ps)))
4		simpl 346	. . . . . . 7 |- ((ph /\ ps) -> ph)
5	4	imim2i 11	. . . . . 6 |- ((x = y -> (ph /\ ps)) -> (x = y -> ph))
6	5	alimi 1338	. . . . 5 |- (A.x(x = y -> (ph /\ ps)) -> A.x(x = y -> ph))
7		sb2 1541	. . . . 5 |- (A.x(x = y -> ph) -> [y / x]ph)
8	3, 6, 7	3syl 24	. . . 4 |- (A.x((ph /\ ps) <-> x = y) -> [y / x]ph)
9		iotaval 5096	. . . . . 6 |- (A.x((ph /\ ps) <-> x = y) -> (iotax(ph /\ ps)) = y)
10	9	eqcomd 1889	. . . . 5 |- (A.x((ph /\ ps) <-> x = y) -> y = (iotax(ph /\ ps)))
11		dfsbcq 2455	. . . . 5 |- (y = (iotax(ph /\ ps)) -> ([y / x]ph <-> [(iotax(ph /\ ps)) / x]ph))
12	10, 11	syl 12	. . . 4 |- (A.x((ph /\ ps) <-> x = y) -> ([y / x]ph <-> [(iotax(ph /\ ps)) / x]ph))
13	8, 12	mpbid 212	. . 3 |- (A.x((ph /\ ps) <-> x = y) -> [(iotax(ph /\ ps)) / x]ph)
14	13	19.23aiv 1674	. 2 |- (E.yA.x((ph /\ ps) <-> x = y) -> [(iotax(ph /\ ps)) / x]ph)
15	1, 14	sylbi 216	1 |- (E!x(ph /\ ps) -> [(iotax(ph /\ ps)) / x]ph)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326  [wsbc 1534  E!weu 1771  iotacio 5087

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-sbc 2454  df-un 2600  df-sn 3049  df-pr 3050  df-uni 3178  df-iota 5089

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