Theorem u4lemnaa 625

Description: Lemma for non-tollens implication study.

Assertion

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

Proof of Theorem u4lemnaa

Step	Hyp	Ref	Expression
1		anor2 81	. 2 ((a →4 b)⊥ ∩ a) = ((a →4 b) ∪ a⊥ )⊥
2		u4lemona 610	. . . 4 ((a →4 b) ∪ a⊥ ) = (a⊥ ∪ b)
3	2	ax-r4 36	. . 3 ((a →4 b) ∪ a⊥ )⊥ = (a⊥ ∪ b)⊥
4		anor1 80	. . . 4 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
5	4	ax-r1 34	. . 3 (a⊥ ∪ b)⊥ = (a ∩ b⊥ )
6	3, 5	ax-r2 35	. 2 ((a →4 b) ∪ a⊥ )⊥ = (a ∩ b⊥ )
7	1, 6	ax-r2 35	1 ((a →4 b)⊥ ∩ a) = (a ∩ b⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 16

This theorem is referenced by:  u4lem1 719

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-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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