Theorem govar2 897

Description: Lemma for converting n-variable to 2n-variable Godowski equations. (Contributed by NM, 19-Nov-1999.)

Hypotheses

Ref	Expression
govar.1	a ≤ b⊥
govar.2	b ≤ c⊥

Assertion

Ref	Expression
govar2	(a ∪ b) ≤ (c →2 a)

Proof of Theorem govar2

Step	Hyp	Ref	Expression
1		govar.2	. . . 4 b ≤ c⊥
2		govar.1	. . . . 5 a ≤ b⊥
3	2	lecon3 157	. . . 4 b ≤ a⊥
4	1, 3	ler2an 173	. . 3 b ≤ (c⊥ ∩ a⊥ )
5	4	lelor 166	. 2 (a ∪ b) ≤ (a ∪ (c⊥ ∩ a⊥ ))
6		df-i2 45	. . 3 (c →2 a) = (a ∪ (c⊥ ∩ a⊥ ))
7	6	ax-r1 35	. 2 (a ∪ (c⊥ ∩ a⊥ )) = (c →2 a)
8	5, 7	lbtr 139	1 (a ∪ b) ≤ (c →2 a)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  gon2n  898  go2n4  899  go2n6  901