Theorem lem3.3.7i5e2 1073

Description: Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to 5, and this is the second part of the equation. (Contributed by Roy F. Longton, 28-Jun-2005.) (Revised by Roy F. Longton, 3-Jul-2005.)

Assertion

Ref	Expression
lem3.3.7i5e2	(a ≡5 (a ∩ b)) = ((a ∩ b) ≡5 a)

Proof of Theorem lem3.3.7i5e2

Step	Hyp	Ref	Expression
1		ancom 74	. . . 4 ((a ∩ b) ∩ a) = (a ∩ (a ∩ b))
2		ancom 74	. . . 4 ((a ∩ b)⊥ ∩ a⊥ ) = (a⊥ ∩ (a ∩ b)⊥ )
3	1, 2	2or 72	. . 3 (((a ∩ b) ∩ a) ∪ ((a ∩ b)⊥ ∩ a⊥ )) = ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
4	3	ax-r1 35	. 2 ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ )) = (((a ∩ b) ∩ a) ∪ ((a ∩ b)⊥ ∩ a⊥ ))
5		df-id5 1047	. 2 (a ≡5 (a ∩ b)) = ((a ∩ (a ∩ b)) ∪ (a⊥ ∩ (a ∩ b)⊥ ))
6		df-id5 1047	. 2 ((a ∩ b) ≡5 a) = (((a ∩ b) ∩ a) ∪ ((a ∩ b)⊥ ∩ a⊥ ))
7	4, 5, 6	3tr1 63	1 (a ≡5 (a ∩ b)) = ((a ∩ b) ≡5 a)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   ≡₅ wid5 22

This theorem was proved from axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-id5 1047

This theorem is referenced by: (None)