Theorem u4lem5n 766

Description: Lemma for unified implication study. (Contributed by NM, 20-Dec-1997.)

Assertion

Ref	Expression
u4lem5n	(a →4 (a →4 b))⊥ = ((a ∪ b) ∩ b⊥ )

Proof of Theorem u4lem5n

Step	Hyp	Ref	Expression
1		u4lem5 764	. . . 4 (a →4 (a →4 b)) = ((a⊥ ∩ b⊥ ) ∪ b)
2		anor3 90	. . . . 5 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
3	2	ax-r5 38	. . . 4 ((a⊥ ∩ b⊥ ) ∪ b) = ((a ∪ b)⊥ ∪ b)
4	1, 3	ax-r2 36	. . 3 (a →4 (a →4 b)) = ((a ∪ b)⊥ ∪ b)
5		oran2 92	. . 3 ((a ∪ b)⊥ ∪ b) = ((a ∪ b) ∩ b⊥ )⊥
6	4, 5	ax-r2 36	. 2 (a →4 (a →4 b)) = ((a ∪ b) ∩ b⊥ )⊥
7	6	con2 67	1 (a →4 (a →4 b))⊥ = ((a ∪ b) ∩ 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:  u4lem6  768