Theorem dfif4 3956

Description: Alternate definition of the conditional operator df-if 3942. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)

Hypothesis

Ref	Expression
dfif3.1

Assertion

Ref	Expression
dfif4

Distinct variable group:   ,

Proof of Theorem dfif4

Step	Hyp	Ref	Expression
1		dfif3.1	. . 3
2	1	dfif3 3955	. 2
3		undir 3746	. 2
4		undi 3744	. . . 4
5		undi 3744	. . . . 5
6		uncom 3647	. . . . . 6
7		unvdif 3902	. . . . . 6
8	6, 7	ineq12i 3697	. . . . 5
9		inv1 3812	. . . . 5
10	5, 8, 9	3eqtri 2490	. . . 4
11	4, 10	ineq12i 3697	. . 3
12		inass 3707	. . 3
13	11, 12	eqtri 2486	. 2
14	2, 3, 13	3eqtri 2490	1

Syntax hints:  =wceq 1395  {cab 2442  cvv 3109  \cdif 3472  u.cun 3473  i^icin 3474  ifcif 3941

This theorem is referenced by:  dfif5  3957

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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942