Theorem u1lemc2 686

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

Hypotheses

Ref	Expression
ulemc2.1	a C b
ulemc2.2	a C c

Assertion

Ref	Expression
u1lemc2	a C (b →1 c)

Proof of Theorem u1lemc2

Step	Hyp	Ref	Expression
1		ulemc2.1	. . . 4 a C b
2	1	comcom2 183	. . 3 a C b⊥
3		ulemc2.2	. . . 4 a C c
4	1, 3	com2an 484	. . 3 a C (b ∩ c)
5	2, 4	com2or 483	. 2 a C (b⊥ ∪ (b ∩ c))
6		df-i1 44	. . 3 (b →1 c) = (b⊥ ∪ (b ∩ c))
7	6	ax-r1 35	. 2 (b⊥ ∪ (b ∩ c)) = (b →1 c)
8	5, 7	cbtr 182	1 a C (b →1 c)

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:  u1lemc3  691