Theorem i3ran 535

Description: WQL (Weak Quantum Logic) rule.

Hypothesis

Ref	Expression
i3ran.1	(a →3 b) = 1

Assertion

Ref	Expression
i3ran	((a ∩ c) →3 (b ∩ c)) = 1

Proof of Theorem i3ran

Step	Hyp	Ref	Expression
1		i3ran.1	. . . . 5 (a →3 b) = 1
2	1	binr1 517	. . . 4 (b⊥ →3 a⊥ ) = 1
3	2	i3ror 532	. . 3 ((b⊥ ∪ c⊥ ) →3 (a⊥ ∪ c⊥ )) = 1
4	3	binr1 517	. 2 ((a⊥ ∪ c⊥ )⊥ →3 (b⊥ ∪ c⊥ )⊥ ) = 1
5		df-a 40	. 2 (a ∩ c) = (a⊥ ∪ c⊥ )⊥
6		df-a 40	. 2 (b ∩ c) = (b⊥ ∪ c⊥ )⊥
7	4, 5, 6	i33tr1 529	1 ((a ∩ c) →3 (b ∩ c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ 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:  i3lan  536  i32an  537