Theorem u3lemnaa 624

Description: Lemma for Kalmbach implication study.

Assertion

Ref	Expression
u3lemnaa	((a →3 b)⊥ ∩ a) = (a ∩ b⊥ )

Proof of Theorem u3lemnaa

Step	Hyp	Ref	Expression
1		anor2 81	. 2 ((a →3 b)⊥ ∩ a) = ((a →3 b) ∪ a⊥ )⊥
2		anor1 80	. . . 4 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
3		u3lemona 609	. . . . . 6 ((a →3 b) ∪ a⊥ ) = (a⊥ ∪ b)
4	3	ax-r4 36	. . . . 5 ((a →3 b) ∪ a⊥ )⊥ = (a⊥ ∪ b)⊥
5	4	ax-r1 34	. . . 4 (a⊥ ∪ b)⊥ = ((a →3 b) ∪ a⊥ )⊥
6	2, 5	ax-r2 35	. . 3 (a ∩ b⊥ ) = ((a →3 b) ∪ a⊥ )⊥
7	6	ax-r1 34	. 2 ((a →3 b) ∪ a⊥ )⊥ = (a ∩ b⊥ )
8	1, 7	ax-r2 35	1 ((a →3 b)⊥ ∩ a) = (a ∩ b⊥ )

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

This theorem is referenced by:  u3lem13a 771  u3lem13b 772

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  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

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