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) ∪ (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) ∪ (1⊥ ∩ a⊥ )) = ((a ∩ 1) ∪ (a⊥ ∩ 0))
9		an1 98	. . . . 5 (a ∩ 1) = a
10		an0 100	. . . . 5 (a⊥ ∩ 0) = 0
11	9, 10	2or 67	. . . 4 ((a ∩ 1) ∪ (a⊥ ∩ 0)) = (a ∪ 0)
12	8, 11	ax-r2 35	. . 3 ((1 ∩ a) ∪ (1⊥ ∩ a⊥ )) = (a ∪ 0)
13		or0 94	. . 3 (a ∪ 0) = a
14	12, 13	ax-r2 35	. 2 ((1 ∩ a) ∪ (1⊥ ∩ a⊥ )) = a
15	1, 14	ax-r2 35	1 (1 ≡ a) = a

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ 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

metamath.org