Theorem u4lem5n 748

Description: Lemma for unified implication study.

Assertion

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

Proof of Theorem u4lem5n

Step	Hyp	Ref	Expression
1		u4lem5 746	. . . 4 (a →4 (a →4 b)) = ((a⊥ ∩ b⊥ ) ∪ b)
2		anor3 82	. . . . 5 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
3	2	ax-r5 37	. . . 4 ((a⊥ ∩ b⊥ ) ∪ b) = ((a ∪ b)⊥ ∪ b)
4	1, 3	ax-r2 35	. . 3 (a →4 (a →4 b)) = ((a ∪ b)⊥ ∪ b)
5		oran2 84	. . 3 ((a ∪ b)⊥ ∪ b) = ((a ∪ b) ∩ b⊥ )⊥
6	4, 5	ax-r2 35	. 2 (a →4 (a →4 b)) = ((a ∪ b) ∩ b⊥ )⊥
7	6	con2 64	1 (a →4 (a →4 b))⊥ = ((a ∪ b) ∩ b⊥ )

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

This theorem is referenced by:  u4lem6 750

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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