Theorem lem3.3.3 1052

Description: Equation 3.3 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton, 27-Jun-2005.) (Revised by Roy F. Longton, 3-Jul-2005.)

Assertion

Ref	Expression
lem3.3.3	((a ≡5 b) →0 (a ↔1 b)) = 1

Proof of Theorem lem3.3.3

Step	Hyp	Ref	Expression
1		df-i0 43	. 2 ((a ≡5 b) →0 (a ↔1 b)) = ((a ≡5 b)⊥ ∪ (a ↔1 b))
2		df-b1 1048	. . 3 (a ↔1 b) = ((a →1 b) ∩ (b →1 a))
3	2	lor 70	. 2 ((a ≡5 b)⊥ ∪ (a ↔1 b)) = ((a ≡5 b)⊥ ∪ ((a →1 b) ∩ (b →1 a)))
4		lem3.3.3lem3 1051	. . 3 (a ≡5 b) ≤ ((a →1 b) ∩ (b →1 a))
5	4	sklem 230	. 2 ((a ≡5 b)⊥ ∪ ((a →1 b) ∩ (b →1 a))) = 1
6	1, 3, 5	3tr 65	1 ((a ≡5 b) →0 (a ↔1 b)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₀ wi0 11   →₁ wi1 12   ≡₅ wid5 22   ↔₁ wb1 24

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

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-le1 130  df-le2 131  df-id5 1047  df-b1 1048

This theorem is referenced by:  lem3.3.5  1055