Theorem u5lemnanb 641

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemnanb	((a →5 b)⊥ ∩ b⊥ ) = ((a ∪ b) ∩ b⊥ )

Proof of Theorem u5lemnanb

Step	Hyp	Ref	Expression
1		u5lemob 616	. . . 4 ((a →5 b) ∪ b) = ((a⊥ ∩ b⊥ ) ∪ b)
2		anor3 82	. . . . 5 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
3	2	ax-r5 37	. . . 4 ((a⊥ ∩ b⊥ ) ∪ b) = ((a ∪ b)⊥ ∪ b)
4	1, 3	ax-r2 35	. . 3 ((a →5 b) ∪ b) = ((a ∪ b)⊥ ∪ b)
5		oran 79	. . 3 ((a →5 b) ∪ b) = ((a →5 b)⊥ ∩ b⊥ )⊥
6		oran2 84	. . 3 ((a ∪ b)⊥ ∪ b) = ((a ∪ b) ∩ b⊥ )⊥
7	4, 5, 6	3tr2 61	. 2 ((a →5 b)⊥ ∩ b⊥ )⊥ = ((a ∪ b) ∩ b⊥ )⊥
8	7	con1 63	1 ((a →5 b)⊥ ∩ b⊥ ) = ((a ∪ b) ∩ b⊥ )

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

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123

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