Theorem u5lem3 735

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u5lem3	(a →5 (b →5 a)) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))

Proof of Theorem u5lem3

Step	Hyp	Ref	Expression
1		u5lemc1b 667	. . 3 a C (b →5 a)
2	1	u5lemc4 687	. 2 (a →5 (b →5 a)) = (a⊥ ∪ (b →5 a))
3		ax-a2 30	. . 3 (a⊥ ∪ (b →5 a)) = ((b →5 a) ∪ a⊥ )
4		u5lemonb 621	. . . 4 ((b →5 a) ∪ a⊥ ) = (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ a⊥ )
5		ancom 68	. . . . . . 7 (b ∩ a) = (a ∩ b)
6		ancom 68	. . . . . . 7 (b⊥ ∩ a) = (a ∩ b⊥ )
7	5, 6	2or 67	. . . . . 6 ((b ∩ a) ∪ (b⊥ ∩ a)) = ((a ∩ b) ∪ (a ∩ b⊥ ))
8	7	ax-r5 37	. . . . 5 (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ a⊥ ) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ )
9		ax-a2 30	. . . . 5 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
10	8, 9	ax-r2 35	. . . 4 (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ a⊥ ) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
11	4, 10	ax-r2 35	. . 3 ((b →5 a) ∪ a⊥ ) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
12	3, 11	ax-r2 35	. 2 (a⊥ ∪ (b →5 a)) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
13	2, 12	ax-r2 35	1 (a →5 (b →5 a)) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 17

This theorem is referenced by:  u5lem3n 738  u5lem4 742

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-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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