Theorem i3lan 536

Description: WQL (Weak Quantum Logic) rule. (Contributed by NM, 7-Nov-1997.)

Hypothesis

Ref	Expression
i3lan.1	(a →3 b) = 1

Assertion

Ref	Expression
i3lan	((c ∩ a) →3 (c ∩ b)) = 1

Proof of Theorem i3lan

Step	Hyp	Ref	Expression
1		i3lan.1	. . 3 (a →3 b) = 1
2	1	i3ran 535	. 2 ((a ∩ c) →3 (b ∩ c)) = 1
3		ancom 74	. 2 (c ∩ a) = (a ∩ c)
4		ancom 74	. 2 (c ∩ b) = (b ∩ c)
5	2, 3, 4	i33tr1 529	1 ((c ∩ a) →3 (c ∩ b)) = 1

Syntax hints:   = wb 1   ∩ wa 7  1wt 8   →₃ wi3 14

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

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  i32an  537