Theorem u5lemnoa 646

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemnoa	((a →5 b)⊥ ∪ a) = ((a ∪ b) ∩ (a ∪ b⊥ ))

Proof of Theorem u5lemnoa

Step	Hyp	Ref	Expression
1		u5lemana 591	. . . 4 ((a →5 b) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
2		ax-a2 30	. . . . 5 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b))
3		anor3 82	. . . . . 6 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
4		anor2 81	. . . . . 6 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
5	3, 4	2or 67	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b)) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )
6	2, 5	ax-r2 35	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )
7	1, 6	ax-r2 35	. . 3 ((a →5 b) ∩ a⊥ ) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )
8		anor1 80	. . 3 ((a →5 b) ∩ a⊥ ) = ((a →5 b)⊥ ∪ a)⊥
9		oran3 85	. . 3 ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ ) = ((a ∪ b) ∩ (a ∪ b⊥ ))⊥
10	7, 8, 9	3tr2 61	. 2 ((a →5 b)⊥ ∪ a)⊥ = ((a ∪ b) ∩ (a ∪ b⊥ ))⊥
11	10	con1 63	1 ((a →5 b)⊥ ∪ a) = ((a ∪ b) ∩ (a ∪ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 17

This theorem is referenced by:  u5lem1 720

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-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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