Theorem u5lemc1 684

Description: Commutation theorem for relevance implication. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u5lemc1	a C (a →5 b)

Proof of Theorem u5lemc1

Step	Hyp	Ref	Expression
1		comanr1 464	. . . 4 a C (a ∩ b)
2		comanr1 464	. . . . 5 a⊥ C (a⊥ ∩ b)
3	2	comcom6 459	. . . 4 a C (a⊥ ∩ b)
4	1, 3	com2or 483	. . 3 a C ((a ∩ b) ∪ (a⊥ ∩ b))
5		comanr1 464	. . . 4 a⊥ C (a⊥ ∩ b⊥ )
6	5	comcom6 459	. . 3 a C (a⊥ ∩ b⊥ )
7	4, 6	com2or 483	. 2 a C (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
8		df-i5 48	. . 3 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
9	8	ax-r1 35	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = (a →5 b)
10	7, 9	cbtr 182	1 a C (a →5 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 16

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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u5lemc5  700  u5lembi  725  u5lem1  738  u5lem4  760  u5lem5  765  u5lem6  769