Theorem i3or 497

Description: Kalmbach implication OR builder. (Contributed by NM, 26-Dec-1997.)

Assertion

Ref	Expression
i3or	((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))) = 1

Proof of Theorem i3or

Step	Hyp	Ref	Expression
1		le1 146	. 2 ((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))) ≤ 1
2		ka4ot 435	. . . 4 ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) = 1
3	2	ax-r1 35	. . 3 1 = ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
4		i3bi 496	. . . . . 6 (((a ∪ c) →3 (b ∪ c)) ∩ ((b ∪ c) →3 (a ∪ c))) = ((a ∪ c) ≡ (b ∪ c))
5	4	ax-r1 35	. . . . 5 ((a ∪ c) ≡ (b ∪ c)) = (((a ∪ c) →3 (b ∪ c)) ∩ ((b ∪ c) →3 (a ∪ c)))
6		lea 160	. . . . 5 (((a ∪ c) →3 (b ∪ c)) ∩ ((b ∪ c) →3 (a ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))
7	5, 6	bltr 138	. . . 4 ((a ∪ c) ≡ (b ∪ c)) ≤ ((a ∪ c) →3 (b ∪ c))
8	7	lelor 166	. . 3 ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))
9	3, 8	bltr 138	. 2 1 ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))
10	1, 9	lebi 145	1 ((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →₃ wi3 14

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-wom 361  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-i3 46  df-le 129  df-le1 130  df-le2 131  df-c1 132  df-c2 133  df-cmtr 134

This theorem is referenced by: (None)