Theorem u1lemc1 662

Description: Commutation theorem for Sasaki implication.

Assertion

Ref	Expression
u1lemc1	a C (a →1 b)

Proof of Theorem u1lemc1

Step	Hyp	Ref	Expression
1		comid 179	. . . 4 a C a
2	1	comcom2 175	. . 3 a C a⊥
3		comanr1 446	. . 3 a C (a ∩ b)
4	2, 3	com2or 465	. 2 a C (a⊥ ∪ (a ∩ b))
5		df-i1 43	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
6	5	ax-r1 34	. 2 (a⊥ ∪ (a ∩ b)) = (a →1 b)
7	4, 6	cbtr 174	1 a C (a →1 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  u1lemc5 678  u12lembi 708  u1lem1 716  u1lem4 739  oas 905  oau 909

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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