Theorem wcon2 208

Description: Weak contraposition. (Contributed by NM, 24-Sep-1997.)

Hypothesis

Ref	Expression
wcon2.1	(a ≡ b⊥ ) = 1

Assertion

Ref	Expression
wcon2	(a⊥ ≡ b) = 1

Proof of Theorem wcon2

Step	Hyp	Ref	Expression
1		conb 122	. . 3 (a⊥ ≡ b) = (a⊥ ⊥ ≡ b⊥ )
2		ax-a1 30	. . . . 5 a = a⊥ ⊥
3	2	rbi 98	. . . 4 (a ≡ b⊥ ) = (a⊥ ⊥ ≡ b⊥ )
4	3	ax-r1 35	. . 3 (a⊥ ⊥ ≡ b⊥ ) = (a ≡ b⊥ )
5	1, 4	ax-r2 36	. 2 (a⊥ ≡ b) = (a ≡ b⊥ )
6		wcon2.1	. 2 (a ≡ b⊥ ) = 1
7	5, 6	ax-r2 36	1 (a⊥ ≡ b) = 1

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ≡ tb 5  1wt 8

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-b 39  df-a 40

This theorem is used by:  wcomlem  382  wcomd  418  wcomdr  421  wcom3i  422  wfh1  423  wfh2  424