Theorem u5lemc1 666

Description: Commutation theorem for relevance implication.

Assertion

Ref	Expression
u5lemc1	a C (a →5 b)

Proof of Theorem u5lemc1

Step	Hyp	Ref	Expression
1		comanr1 446	. . . 4 a C (a ∩ b)
2		comanr1 446	. . . . 5 a⊥ C (a⊥ ∩ b)
3	2	comcom6 441	. . . 4 a C (a⊥ ∩ b)
4	1, 3	com2or 465	. . 3 a C ((a ∩ b) ∪ (a⊥ ∩ b))
5		comanr1 446	. . . 4 a⊥ C (a⊥ ∩ b⊥ )
6	5	comcom6 441	. . 3 a C (a⊥ ∩ b⊥ )
7	4, 6	com2or 465	. 2 a C (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
8		df-i5 47	. . 3 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
9	8	ax-r1 34	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = (a →5 b)
10	7, 9	cbtr 174	1 a C (a →5 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 17

This theorem is referenced by:  u5lemc5 682  u5lembi 707  u5lem1 720  u5lem4 742  u5lem5 747  u5lem6 751

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-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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