Theorem i3or 479

Description: Kalmbach implication OR builder.

Assertion

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

Proof of Theorem i3or

Step	Hyp	Ref	Expression
1		le1 138	. 2 ((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))) ≤ 1
2		ka4ot 417	. . . 4 ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) = 1
3	2	ax-r1 34	. . 3 1 = ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c)))
4		i3bi 478	. . . . . 6 (((a ∪ c) →3 (b ∪ c)) ∩ ((b ∪ c) →3 (a ∪ c))) = ((a ∪ c) ≡ (b ∪ c))
5	4	ax-r1 34	. . . . 5 ((a ∪ c) ≡ (b ∪ c)) = (((a ∪ c) →3 (b ∪ c)) ∩ ((b ∪ c) →3 (a ∪ c)))
6		lea 152	. . . . 5 (((a ∪ c) →3 (b ∪ c)) ∩ ((b ∪ c) →3 (a ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))
7	5, 6	bltr 130	. . . 4 ((a ∪ c) ≡ (b ∪ c)) ≤ ((a ∪ c) →3 (b ∪ c))
8	7	lelor 158	. . 3 ((a ≡ b)⊥ ∪ ((a ∪ c) ≡ (b ∪ c))) ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))
9	3, 8	bltr 130	. 2 1 ≤ ((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c)))
10	1, 9	lebi 137	1 ((a ≡ b)⊥ ∪ ((a ∪ c) →3 (b ∪ c))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9   →₃ wi3 15

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-wom 343  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-i3 45  df-le 121  df-le1 122  df-le2 123  df-c1 124  df-c2 125  df-cmtr 126

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