Theorem i3ran 517

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 499	. . . 4 (b⊥ →3 a⊥ ) = 1
3	2	i3ror 514	. . 3 ((b⊥ ∪ c⊥ ) →3 (a⊥ ∪ c⊥ )) = 1
4	3	binr1 499	. 2 ((a⊥ ∪ c⊥ )⊥ →3 (b⊥ ∪ c⊥ )⊥ ) = 1
5		df-a 39	. 2 (a ∩ c) = (a⊥ ∪ c⊥ )⊥
6		df-a 39	. 2 (b ∩ c) = (b⊥ ∪ c⊥ )⊥
7	4, 5, 6	i33tr1 511	1 ((a ∩ c) →3 (b ∩ c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₃ wi3 15

This theorem is referenced by:  i3lan 518  i32an 519

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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