Theorem iota4 5574

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

Assertion

Ref	Expression
iota4

Proof of Theorem iota4

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2286	. 2
2		bi2 198	. . . . . 6
3	2	alimi 1633	. . . . 5
4		sb2 2093	. . . . 5
5	3, 4	syl 16	. . . 4
6		iotaval 5567	. . . . . 6
7	6	eqcomd 2465	. . . . 5
8		dfsbcq2 3330	. . . . 5
9	7, 8	syl 16	. . . 4
10	5, 9	mpbid 210	. . 3
11	10	exlimiv 1722	. 2
12	1, 11	sylbi 195	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  E.wex 1612  [wsb 1739  E!weu 2282  [.wsbc 3327  iotacio 5554

This theorem is referenced by:  iota4an  5575  iotacl  5579  pm14.24  31339  sbiota1  31341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-v 3111  df-sbc 3328  df-un 3480  df-sn 4030  df-pr 4032  df-uni 4250  df-iota 5556