Theorem u1lemc1 680

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

Assertion

Ref	Expression
u1lemc1	a C (a →1 b)

Proof of Theorem u1lemc1

Step	Hyp	Ref	Expression
1		comid 187	. . . 4 a C a
2	1	comcom2 183	. . 3 a C a⊥
3		comanr1 464	. . 3 a C (a ∩ b)
4	2, 3	com2or 483	. 2 a C (a⊥ ∪ (a ∩ b))
5		df-i1 44	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
6	5	ax-r1 35	. 2 (a⊥ ∪ (a ∩ b)) = (a →1 b)
7	4, 6	cbtr 182	1 a C (a →1 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u1lemc5  696  u12lembi  726  u1lem1  734  u1lem4  757  oas  925  oau  929