Theorem u3lemnaa 642

Description: Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

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

Proof of Theorem u3lemnaa

Step	Hyp	Ref	Expression
1		anor2 89	. 2 ((a →3 b)⊥ ∩ a) = ((a →3 b) ∪ a⊥ )⊥
2		anor1 88	. . . 4 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
3		u3lemona 627	. . . . . 6 ((a →3 b) ∪ a⊥ ) = (a⊥ ∪ b)
4	3	ax-r4 37	. . . . 5 ((a →3 b) ∪ a⊥ )⊥ = (a⊥ ∪ b)⊥
5	4	ax-r1 35	. . . 4 (a⊥ ∪ b)⊥ = ((a →3 b) ∪ a⊥ )⊥
6	2, 5	ax-r2 36	. . 3 (a ∩ b⊥ ) = ((a →3 b) ∪ a⊥ )⊥
7	6	ax-r1 35	. 2 ((a →3 b) ∪ a⊥ )⊥ = (a ∩ b⊥ )
8	1, 7	ax-r2 36	1 ((a →3 b)⊥ ∩ a) = (a ∩ b⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131

This theorem is referenced by:  u3lem13a  789  u3lem13b  790