Theorem pm14.24 16397

Description: Theorem *14.24 in [WhiteheadRussell] p. 191.

Assertion

Ref	Expression
pm14.24	|- (E!xph -> A.y([y / x]ph <-> y = (iotaxph)))

Distinct variable groups:   x,y   ph,y

Proof of Theorem pm14.24

Step	Hyp	Ref	Expression
1		hbeu1 1781	. . . . 5 |- (E!xph -> A.xE!xph)
2		hbs1 1722	. . . . 5 |- ([y / x]ph -> A.x[y / x]ph)
3		pm14.12 16385	. . . . . . . . . 10 |- (E!xph -> A.xA.y((ph /\ [y / x]ph) -> x = y))
4	3	19.21bbi 1409	. . . . . . . . 9 |- (E!xph -> ((ph /\ [y / x]ph) -> x = y))
5	4	ancomsd 485	. . . . . . . 8 |- (E!xph -> (([y / x]ph /\ ph) -> x = y))
6	5	expdimp 406	. . . . . . 7 |- ((E!xph /\ [y / x]ph) -> (ph -> x = y))
7		sbequ2 1543	. . . . . . . . 9 |- (x = y -> ([y / x]ph -> ph))
8	7	com12 14	. . . . . . . 8 |- ([y / x]ph -> (x = y -> ph))
9	8	adantl 424	. . . . . . 7 |- ((E!xph /\ [y / x]ph) -> (x = y -> ph))
10	6, 9	impbid 574	. . . . . 6 |- ((E!xph /\ [y / x]ph) -> (ph <-> x = y))
11	10	ex 402	. . . . 5 |- (E!xph -> ([y / x]ph -> (ph <-> x = y)))
12	1, 2, 11	19.21ad 1406	. . . 4 |- (E!xph -> ([y / x]ph -> A.x(ph <-> x = y)))
13		iotaval 5096	. . . . 5 |- (A.x(ph <-> x = y) -> (iotaxph) = y)
14	13	eqcomd 1889	. . . 4 |- (A.x(ph <-> x = y) -> y = (iotaxph))
15	12, 14	syl6 25	. . 3 |- (E!xph -> ([y / x]ph -> y = (iotaxph)))
16		dfsbcq 2455	. . . 4 |- (y = (iotaxph) -> ([y / x]ph <-> [(iotaxph) / x]ph))
17		iota4 5100	. . . 4 |- (E!xph -> [(iotaxph) / x]ph)
18	16, 17	syl5cbir 228	. . 3 |- (E!xph -> (y = (iotaxph) -> [y / x]ph))
19	15, 18	impbid 574	. 2 |- (E!xph -> ([y / x]ph <-> y = (iotaxph)))
20	19	19.21aiv 1664	1 |- (E!xph -> A.y([y / x]ph <-> y = (iotaxph)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  [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-mo 1776  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

Copyright terms: Public domain