Theorem con3d 152

Description: A contraposition deduction.

Hypothesis

Ref	Expression
con3d.1	|- (R, S) |= T

Assertion

Ref	Expression
con3d	|- (R, (~ T)) |= (~ S)

Proof of Theorem con3d

Step	Hyp	Ref	Expression
1		wnot 128	. . 3 |- ~ :(* -> *)
2		con3d.1	. . . 4 |- (R, S) |= T
3	2	ax-cb2 30	. . 3 |- T:*
4	1, 3	wc 45	. 2 |- (~ T):*
5	3	notnot1 150	. . 3 |- T |= (~ (~ T))
6	2, 5	syl 16	. 2 |- (R, S) |= (~ (~ T))
7	4, 6	con2d 151	1 |- (R, (~ T)) |= (~ S)

Syntax hints:  *hb 3  kc 5  kct 10   |= wffMMJ2 11  ~ tne 110

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103

This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120

This theorem is referenced by:  alnex  174