Theorem u2lembi 703

Description: Dishkant implication and biconditional.

Assertion

Ref	Expression
u2lembi	((a →2 b) ∩ (b →2 a)) = (a ≡ b)

Proof of Theorem u2lembi

Step	Hyp	Ref	Expression
1		ancom 68	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ (a⊥ ∩ b⊥ ))) = ((a ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (a⊥ ∩ b⊥ )))
2		coman1 177	. . . . . 6 (a⊥ ∩ b⊥ ) C a⊥
3	2	comcom7 442	. . . . 5 (a⊥ ∩ b⊥ ) C a
4		coman2 178	. . . . . 6 (a⊥ ∩ b⊥ ) C b⊥
5	4	comcom7 442	. . . . 5 (a⊥ ∩ b⊥ ) C b
6	3, 5	fh3r 457	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (a⊥ ∩ b⊥ )))
7	6	ax-r1 34	. . 3 ((a ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (a⊥ ∩ b⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
8	1, 7	ax-r2 35	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ (a⊥ ∩ b⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
9		df-i2 44	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
10		df-i2 44	. . . 4 (b →2 a) = (a ∪ (b⊥ ∩ a⊥ ))
11		ancom 68	. . . . 5 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
12	11	lor 66	. . . 4 (a ∪ (b⊥ ∩ a⊥ )) = (a ∪ (a⊥ ∩ b⊥ ))
13	10, 12	ax-r2 35	. . 3 (b →2 a) = (a ∪ (a⊥ ∩ b⊥ ))
14	9, 13	2an 72	. 2 ((a →2 b) ∩ (b →2 a)) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ (a ∪ (a⊥ ∩ b⊥ )))
15		dfb 86	. 2 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
16	8, 14, 15	3tr1 60	1 ((a →2 b) ∩ (b →2 a)) = (a ≡ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₂ wi2 14

This theorem is referenced by:  i2bi 704  mloa 998

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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