Theorem ud2lem0a 258

Description: Introduce →₂ to the left. (Contributed by NM, 23-Nov-1997.)

Hypothesis

Ref	Expression
ud2lem0a.1	a = b

Assertion

Ref	Expression
ud2lem0a	(c →2 a) = (c →2 b)

Proof of Theorem ud2lem0a

Step	Hyp	Ref	Expression
1		ud2lem0a.1	. . 3 a = b
2	1	ax-r4 37	. . . 4 a⊥ = b⊥
3	2	lan 77	. . 3 (c⊥ ∩ a⊥ ) = (c⊥ ∩ b⊥ )
4	1, 3	2or 72	. 2 (a ∪ (c⊥ ∩ a⊥ )) = (b ∪ (c⊥ ∩ b⊥ ))
5		df-i2 45	. 2 (c →2 a) = (a ∪ (c⊥ ∩ a⊥ ))
6		df-i2 45	. 2 (c →2 b) = (b ∪ (c⊥ ∩ b⊥ ))
7	4, 5, 6	3tr1 63	1 (c →2 a) = (c →2 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i2 45

This theorem is referenced by:  i2i1  267  i1i2con2  269  nom41  326  ud2  596  3vth6  809  2oath1i1  827  1oath1i1u  828