Theorem 4oaiii 1019

Description: Proper OA analog to Godowski/Greechie, Eq. III.

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
4oaiii	((a →1 d) ∩ f) = ((b →1 d) ∩ f)

Proof of Theorem 4oaiii

Step	Hyp	Ref	Expression
1		4oa.1	. . . 4 e = (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))))
2		4oa.2	. . . 4 f = (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e)
3	1, 2	4oa 1018	. . 3 ((a →1 d) ∩ f) ≤ (b →1 d)
4		lear 153	. . 3 ((a →1 d) ∩ f) ≤ f
5	3, 4	ler2an 165	. 2 ((a →1 d) ∩ f) ≤ ((b →1 d) ∩ f)
6		ancom 68	. . . . 5 (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))) = (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))))
7	1, 6	ax-r2 35	. . . 4 e = (((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d))) ∩ ((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))))
8		ancom 68	. . . . . . 7 (a ∩ b) = (b ∩ a)
9		ancom 68	. . . . . . 7 ((a →1 d) ∩ (b →1 d)) = ((b →1 d) ∩ (a →1 d))
10	8, 9	2or 67	. . . . . 6 ((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) = ((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d)))
11	10	ax-r5 37	. . . . 5 (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ e) = (((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ e)
12	2, 11	ax-r2 35	. . . 4 f = (((b ∩ a) ∪ ((b →1 d) ∩ (a →1 d))) ∪ e)
13	7, 12	4oa 1018	. . 3 ((b →1 d) ∩ f) ≤ (a →1 d)
14		lear 153	. . 3 ((b →1 d) ∩ f) ≤ f
15	13, 14	ler2an 165	. 2 ((b →1 d) ∩ f) ≤ ((a →1 d) ∩ f)
16	5, 15	lebi 137	1 ((a →1 d) ∩ f) = ((b →1 d) ∩ f)

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  4oath1 1020

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  ax-4oa 1012

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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