Theorem oi3oa3lem1 732

Description: An attempt at the OA3 conjecture, which is true if (a ≡ b) = 1. (Contributed by Josiah Burroughs, 27-May-2004.)

Hypothesis

Ref	Expression
oi3oa3lem1.1	1 = (b ≡ a)

Assertion

Ref	Expression
oi3oa3lem1	(((a →1 c) ∩ (b →1 c)) ∪ (a ∩ b)) = 1

Proof of Theorem oi3oa3lem1

Step	Hyp	Ref	Expression
1		oi3oa3lem1.1	. . . . . 6 1 = (b ≡ a)
2	1	r3a 440	. . . . 5 b = a
3	2	ud1lem0b 256	. . . 4 (b →1 c) = (a →1 c)
4	3	lan 77	. . 3 ((a →1 c) ∩ (b →1 c)) = ((a →1 c) ∩ (a →1 c))
5	2	lan 77	. . 3 (a ∩ b) = (a ∩ a)
6	4, 5	2or 72	. 2 (((a →1 c) ∩ (b →1 c)) ∪ (a ∩ b)) = (((a →1 c) ∩ (a →1 c)) ∪ (a ∩ a))
7		anidm 111	. . 3 ((a →1 c) ∩ (a →1 c)) = (a →1 c)
8		anidm 111	. . 3 (a ∩ a) = a
9	7, 8	2or 72	. 2 (((a →1 c) ∩ (a →1 c)) ∪ (a ∩ a)) = ((a →1 c) ∪ a)
10		u1lemoa 620	. 2 ((a →1 c) ∪ a) = 1
11	6, 9, 10	3tr 65	1 (((a →1 c) ∩ (b →1 c)) ∪ (a ∩ b)) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44

This theorem is referenced by:  oi3oa3  733