Theorem 1b 109

Description: Identity law.

Assertion

Ref	Expression
1b	(1 == a) = a

Proof of Theorem 1b

Step	Hyp	Ref	Expression
1		dfb 86	. 2 (1 == a) = ((1 ^ a) v (1_|_ ^ a_|_))
2		ancom 68	. . . . 5 (1 ^ a) = (a ^ 1)
3		ancom 68	. . . . . 6 (1_|_ ^ a_|_) = (a_|_ ^ 1_|_)
4		df-f 41	. . . . . . . 8 0 = 1_|_
5	4	ax-r1 34	. . . . . . 7 1_|_ = 0
6	5	lan 70	. . . . . 6 (a_|_ ^ 1_|_) = (a_|_ ^ 0)
7	3, 6	ax-r2 35	. . . . 5 (1_|_ ^ a_|_) = (a_|_ ^ 0)
8	2, 7	2or 67	. . . 4 ((1 ^ a) v (1_|_ ^ a_|_)) = ((a ^ 1) v (a_|_ ^ 0))
9		an1 98	. . . . 5 (a ^ 1) = a
10		an0 100	. . . . 5 (a_|_ ^ 0) = 0
11	9, 10	2or 67	. . . 4 ((a ^ 1) v (a_|_ ^ 0)) = (a v 0)
12	8, 11	ax-r2 35	. . 3 ((1 ^ a) v (1_|_ ^ a_|_)) = (a v 0)
13		or0 94	. . 3 (a v 0) = a
14	12, 13	ax-r2 35	. 2 ((1 ^ a) v (1_|_ ^ a_|_)) = a
15	1, 14	ax-r2 35	1 (1 == a) = a

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9  0wf 10

This theorem is referenced by:  wr3 190  woml6 418  woml7 419  r3b 424

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41

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