Theorem lem3.4.6 1078

Description: Equation 3.14 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton, 3-Jul-05.)

Hypothesis

Ref	Expression
lem3.4.6.1	(a ≡5 b) = 1

Assertion

Ref	Expression
lem3.4.6	((a ∪ c) ≡5 (b ∪ c)) = 1

Proof of Theorem lem3.4.6

Step	Hyp	Ref	Expression
1		lem3.3.6 1055	. . . 4 (a →2 (b ∪ c)) = ((a ∪ c) →2 (b ∪ c))
2	1	ax-r1 35	. . 3 ((a ∪ c) →2 (b ∪ c)) = (a →2 (b ∪ c))
3		lem3.4.6.1	. . . 4 (a ≡5 b) = 1
4	3	lem3.4.5 1077	. . 3 (a →2 (b ∪ c)) = 1
5	2, 4	ax-r2 36	. 2 ((a ∪ c) →2 (b ∪ c)) = 1
6		lem3.3.6 1055	. . . 4 (b →2 (a ∪ c)) = ((b ∪ c) →2 (a ∪ c))
7	6	ax-r1 35	. . 3 ((b ∪ c) →2 (a ∪ c)) = (b →2 (a ∪ c))
8		df-id5 1046	. . . . 5 (b ≡5 a) = ((b ∩ a) ∪ (b⊥ ∩ a⊥ ))
9		ancom 74	. . . . . . 7 (b ∩ a) = (a ∩ b)
10		ancom 74	. . . . . . 7 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
11	9, 10	2or 72	. . . . . 6 ((b ∩ a) ∪ (b⊥ ∩ a⊥ )) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
12		df-id5 1046	. . . . . . 7 (a ≡5 b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
13	12	ax-r1 35	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a ≡5 b)
14	11, 13, 3	3tr 65	. . . . 5 ((b ∩ a) ∪ (b⊥ ∩ a⊥ )) = 1
15	8, 14	ax-r2 36	. . . 4 (b ≡5 a) = 1
16	15	lem3.4.5 1077	. . 3 (b →2 (a ∪ c)) = 1
17	7, 16	ax-r2 36	. 2 ((b ∪ c) →2 (a ∪ c)) = 1
18	5, 17	lem3.4.4 1076	1 ((a ∪ c) ≡5 (b ∪ c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₂ wi2 13   ≡₅ wid5 22

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

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-id5 1046  df-b1 1047

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