Theorem u1lemc2 668

Description: Commutation theorem for Sasaki implication.

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 175	. . 3 a C b⊥
3		ulemc2.2	. . . 4 a C c
4	1, 3	com2an 466	. . 3 a C (b ∩ c)
5	2, 4	com2or 465	. 2 a C (b⊥ ∪ (b ∩ c))
6		df-i1 43	. . 3 (b →1 c) = (b⊥ ∪ (b ∩ c))
7	6	ax-r1 34	. 2 (b⊥ ∪ (b ∩ c)) = (b →1 c)
8	5, 7	cbtr 174	1 a C (b →1 c)

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

This theorem is referenced by:  u1lemc3 673

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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