Theorem con3d 162

Description: A contraposition deduction. (Contributed by Mario Carneiro, 10-Oct-2014.)

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 138	. . 3 |- ~ :(* -> *)
2		con3d.1	. . . 4 |- (R, S) |= T
3	2	ax-cb2 30	. . 3 |- T:*
4	1, 3	wc 50	. 2 |- (~ T):*
5	3	notnot1 160	. . 3 |- T |= (~ (~ T))
6	2, 5	syl 16	. 2 |- (R, S) |= (~ (~ T))
7	4, 6	con2d 161	1 |- (R, (~ T)) |= (~ S)

Syntax hints:  *hb 3  kc 5  kct 10   |= wffMMJ2 11  ~ tne 120

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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130

This theorem is referenced by:  alnex  186