Theorem 4oath1 1041

Description: Proper 4-OA theorem. (Contributed by NM, 29-Dec-1998.)

Hypotheses

Ref	Expression
4oa.1	e = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
4oa.2	f = (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e)

Assertion

Ref	Expression
4oath1	((a →1 d) ∩ f) = ((a →1 d) ∩ (b →1 d))

Proof of Theorem 4oath1

Step	Hyp	Ref	Expression
1		4oa.1	. . . . . 6 e = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
2		4oa.2	. . . . . 6 f = (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e)
3	1, 2	4oaiii 1040	. . . . 5 ((a →1 d) ∩ f) = ((b →1 d) ∩ f)
4	3	lan 77	. . . 4 (((a →1 d) ∩ f) ∩ ((a →1 d) ∩ f)) = (((a →1 d) ∩ f) ∩ ((b →1 d) ∩ f))
5		or32 82	. . . . . . 7 (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e) = (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))
6	2, 5	ax-r2 36	. . . . . 6 f = (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))
7	6	lan 77	. . . . 5 ((a →1 d) ∩ f) = ((a →1 d) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d))))
8	6	lan 77	. . . . 5 ((b →1 d) ∩ f) = ((b →1 d) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d))))
9	7, 8	2an 79	. . . 4 (((a →1 d) ∩ f) ∩ ((b →1 d) ∩ f)) = (((a →1 d) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))) ∩ ((b →1 d) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))))
10	4, 9	ax-r2 36	. . 3 (((a →1 d) ∩ f) ∩ ((a →1 d) ∩ f)) = (((a →1 d) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))) ∩ ((b →1 d) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))))
11		anidm 111	. . . 4 (((a →1 d) ∩ f) ∩ ((a →1 d) ∩ f)) = ((a →1 d) ∩ f)
12	11	ax-r1 35	. . 3 ((a →1 d) ∩ f) = (((a →1 d) ∩ f) ∩ ((a →1 d) ∩ f))
13		anandir 115	. . 3 (((a →1 d) ∩ (b →1 d)) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))) = (((a →1 d) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))) ∩ ((b →1 d) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))))
14	10, 12, 13	3tr1 63	. 2 ((a →1 d) ∩ f) = (((a →1 d) ∩ (b →1 d)) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d))))
15		ax-a2 31	. . 3 (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d))) = (((a →1 d) ∩ (b →1 d)) ∪ ((a ∩ b) ∪ e))
16	15	lan 77	. 2 (((a →1 d) ∩ (b →1 d)) ∩ (((a ∩ b) ∪ e) ∪ ((a →1 d) ∩ (b →1 d)))) = (((a →1 d) ∩ (b →1 d)) ∩ (((a →1 d) ∩ (b →1 d)) ∪ ((a ∩ b) ∪ e)))
17		anabs 121	. 2 (((a →1 d) ∩ (b →1 d)) ∩ (((a →1 d) ∩ (b →1 d)) ∪ ((a ∩ b) ∪ e))) = ((a →1 d) ∩ (b →1 d))
18	14, 16, 17	3tr 65	1 ((a →1 d) ∩ f) = ((a →1 d) ∩ (b →1 d))

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7   →₁ wi1 12

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  ax-4oa 1033

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  4oagen1  1042