Theorem i5con 264

Description: Converse of ->5.

Assertion

Ref	Expression
i5con	(a ->5 b) = (b_|_ ->5 a_|_)

Proof of Theorem i5con

Step	Hyp	Ref	Expression
1		ancom 68	. . . 4 (a_|_ ^ b_|_) = (b_|_ ^ a_|_)
2		ax-a2 30	. . . . 5 ((a ^ b) v (a_|_ ^ b)) = ((a_|_ ^ b) v (a ^ b))
3		ancom 68	. . . . . . 7 (a_|_ ^ b) = (b ^ a_|_)
4		ax-a1 29	. . . . . . . 8 b = b_|__|_
5	4	ran 71	. . . . . . 7 (b ^ a_|_) = (b_|__|_ ^ a_|_)
6	3, 5	ax-r2 35	. . . . . 6 (a_|_ ^ b) = (b_|__|_ ^ a_|_)
7		ancom 68	. . . . . . 7 (a ^ b) = (b ^ a)
8		ax-a1 29	. . . . . . . 8 a = a_|__|_
9	4, 8	2an 72	. . . . . . 7 (b ^ a) = (b_|__|_ ^ a_|__|_)
10	7, 9	ax-r2 35	. . . . . 6 (a ^ b) = (b_|__|_ ^ a_|__|_)
11	6, 10	2or 67	. . . . 5 ((a_|_ ^ b) v (a ^ b)) = ((b_|__|_ ^ a_|_) v (b_|__|_ ^ a_|__|_))
12	2, 11	ax-r2 35	. . . 4 ((a ^ b) v (a_|_ ^ b)) = ((b_|__|_ ^ a_|_) v (b_|__|_ ^ a_|__|_))
13	1, 12	2or 67	. . 3 ((a_|_ ^ b_|_) v ((a ^ b) v (a_|_ ^ b))) = ((b_|_ ^ a_|_) v ((b_|__|_ ^ a_|_) v (b_|__|_ ^ a_|__|_)))
14		ax-a2 30	. . 3 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v ((a ^ b) v (a_|_ ^ b)))
15		ax-a3 31	. . 3 (((b_|_ ^ a_|_) v (b_|__|_ ^ a_|_)) v (b_|__|_ ^ a_|__|_)) = ((b_|_ ^ a_|_) v ((b_|__|_ ^ a_|_) v (b_|__|_ ^ a_|__|_)))
16	13, 14, 15	3tr1 60	. 2 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = (((b_|_ ^ a_|_) v (b_|__|_ ^ a_|_)) v (b_|__|_ ^ a_|__|_))
17		df-i5 47	. 2 (a ->5 b) = (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))
18		df-i5 47	. 2 (b_|_ ->5 a_|_) = (((b_|_ ^ a_|_) v (b_|__|_ ^ a_|_)) v (b_|__|_ ^ a_|__|_))
19	16, 17, 18	3tr1 60	1 (a ->5 b) = (b_|_ ->5 a_|_)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->5 wi5 17

This theorem is referenced by:  nom45 322

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i5 47

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