Theorem u3lemnanb 639

Description: Lemma for Kalmbach implication study.

Assertion

Ref	Expression
u3lemnanb	((a →3 b)⊥ ∩ b⊥ ) = (a ∩ b⊥ )

Proof of Theorem u3lemnanb

Step	Hyp	Ref	Expression
1		u3lemob 614	. . 3 ((a →3 b) ∪ b) = (a⊥ ∪ b)
2		oran 79	. . 3 ((a →3 b) ∪ b) = ((a →3 b)⊥ ∩ b⊥ )⊥
3		oran2 84	. . 3 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
4	1, 2, 3	3tr2 61	. 2 ((a →3 b)⊥ ∩ b⊥ )⊥ = (a ∩ b⊥ )⊥
5	4	con1 63	1 ((a →3 b)⊥ ∩ b⊥ ) = (a ∩ b⊥ )

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

This theorem is referenced by:  u3lem3 733

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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