Theorem u4lemnob 673

Description: Lemma for non-tollens implication study. (Contributed by NM, 16-Dec-1997.)

Assertion

Ref	Expression
u4lemnob	((a →4 b)⊥ ∪ b) = ((a ∩ b⊥ ) ∪ b)

Proof of Theorem u4lemnob

Step	Hyp	Ref	Expression
1		u4lemanb 618	. . . 4 ((a →4 b) ∩ b⊥ ) = ((a⊥ ∪ b) ∩ b⊥ )
2		oran2 92	. . . . 5 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
3	2	ran 78	. . . 4 ((a⊥ ∪ b) ∩ b⊥ ) = ((a ∩ b⊥ )⊥ ∩ b⊥ )
4	1, 3	ax-r2 36	. . 3 ((a →4 b) ∩ b⊥ ) = ((a ∩ b⊥ )⊥ ∩ b⊥ )
5		anor1 88	. . 3 ((a →4 b) ∩ b⊥ ) = ((a →4 b)⊥ ∪ b)⊥
6		anor3 90	. . 3 ((a ∩ b⊥ )⊥ ∩ b⊥ ) = ((a ∩ b⊥ ) ∪ b)⊥
7	4, 5, 6	3tr2 64	. 2 ((a →4 b)⊥ ∪ b)⊥ = ((a ∩ b⊥ ) ∪ b)⊥
8	7	con1 66	1 ((a →4 b)⊥ ∪ b) = ((a ∩ b⊥ ) ∪ b)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 15

This proof depends on 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 proof 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 used by: (None)