Theorem u2lemnaa 641

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

Assertion

Ref	Expression
u2lemnaa	((a →2 b)⊥ ∩ a) = (a ∩ b⊥ )

Proof of Theorem u2lemnaa

Step	Hyp	Ref	Expression
1		anor2 89	. . 3 ((a →2 b)⊥ ∩ a) = ((a →2 b) ∪ a⊥ )⊥
2		u2lemona 626	. . . 4 ((a →2 b) ∪ a⊥ ) = (a⊥ ∪ b)
3	2	ax-r4 37	. . 3 ((a →2 b) ∪ a⊥ )⊥ = (a⊥ ∪ b)⊥
4	1, 3	ax-r2 36	. 2 ((a →2 b)⊥ ∩ a) = (a⊥ ∪ b)⊥
5		anor1 88	. . 3 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
6	5	ax-r1 35	. 2 (a⊥ ∪ b)⊥ = (a ∩ b⊥ )
7	4, 6	ax-r2 36	1 ((a →2 b)⊥ ∩ a) = (a ∩ b⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

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

This theorem depends on definitions:  df-a 40  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  u2lem7  773