Theorem u5lem4 742

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u5lem4	(a →5 (a →5 (b →5 a))) = (a →5 (b →5 a))

Proof of Theorem u5lem4

Step	Hyp	Ref	Expression
1		u5lemc1 666	. . 3 a C (a →5 (b →5 a))
2	1	u5lemc4 687	. 2 (a →5 (a →5 (b →5 a))) = (a⊥ ∪ (a →5 (b →5 a)))
3		u5lem3 735	. . . 4 (a →5 (b →5 a)) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
4	3	lor 66	. . 3 (a⊥ ∪ (a →5 (b →5 a))) = (a⊥ ∪ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))))
5		ax-a3 31	. . . . 5 ((a⊥ ∪ a⊥ ) ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a⊥ ∪ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))))
6	5	ax-r1 34	. . . 4 (a⊥ ∪ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))) = ((a⊥ ∪ a⊥ ) ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
7		oridm 102	. . . . . 6 (a⊥ ∪ a⊥ ) = a⊥
8	7	ax-r5 37	. . . . 5 ((a⊥ ∪ a⊥ ) ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
9	3	ax-r1 34	. . . . 5 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a →5 (b →5 a))
10	8, 9	ax-r2 35	. . . 4 ((a⊥ ∪ a⊥ ) ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a →5 (b →5 a))
11	6, 10	ax-r2 35	. . 3 (a⊥ ∪ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))) = (a →5 (b →5 a))
12	4, 11	ax-r2 35	. 2 (a⊥ ∪ (a →5 (b →5 a))) = (a →5 (b →5 a))
13	2, 12	ax-r2 35	1 (a →5 (a →5 (b →5 a))) = (a →5 (b →5 a))

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

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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