Theorem u4lemnaa 643

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

Assertion

Ref	Expression
u4lemnaa	((a →4 b)⊥ ∩ a) = (a ∩ b⊥ )

Proof of Theorem u4lemnaa

Step	Hyp	Ref	Expression
1		anor2 89	. 2 ((a →4 b)⊥ ∩ a) = ((a →4 b) ∪ a⊥ )⊥
2		u4lemona 628	. . . 4 ((a →4 b) ∪ a⊥ ) = (a⊥ ∪ b)
3	2	ax-r4 37	. . 3 ((a →4 b) ∪ a⊥ )⊥ = (a⊥ ∪ b)⊥
4		anor1 88	. . . 4 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
5	4	ax-r1 35	. . 3 (a⊥ ∪ b)⊥ = (a ∩ b⊥ )
6	3, 5	ax-r2 36	. 2 ((a →4 b) ∪ a⊥ )⊥ = (a ∩ b⊥ )
7	1, 6	ax-r2 36	1 ((a →4 b)⊥ ∩ a) = (a ∩ b⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 15

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u4lem1  737