Theorem con2d 161

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

Hypotheses

Ref	Expression
con2d.1	⊢ T:∗
con2d.2	⊢ (R, S)⊧(¬ T)

Assertion

Ref	Expression
con2d	⊢ (R, T)⊧(¬ S)

Proof of Theorem con2d

Step	Hyp	Ref	Expression
1		con2d.1	. . . . 5 ⊢ T:∗
2		wfal 135	. . . . 5 ⊢ ⊥:∗
3		con2d.2	. . . . . 6 ⊢ (R, S)⊧(¬ T)
4	3	ax-cb1 29	. . . . . . 7 ⊢ (R, S):∗
5	1	notval 145	. . . . . . 7 ⊢ ⊤⊧[(¬ T) = [T ⇒ ⊥]]
6	4, 5	a1i 28	. . . . . 6 ⊢ (R, S)⊧[(¬ T) = [T ⇒ ⊥]]
7	3, 6	mpbi 82	. . . . 5 ⊢ (R, S)⊧[T ⇒ ⊥]
8	1, 2, 7	imp 157	. . . 4 ⊢ ((R, S), T)⊧⊥
9	8	an32s 60	. . 3 ⊢ ((R, T), S)⊧⊥
10	9	ex 158	. 2 ⊢ (R, T)⊧[S ⇒ ⊥]
11	4	wctl 33	. . . 4 ⊢ R:∗
12	11, 1	wct 48	. . 3 ⊢ (R, T):∗
13	4	wctr 34	. . . 4 ⊢ S:∗
14	13	notval 145	. . 3 ⊢ ⊤⊧[(¬ S) = [S ⇒ ⊥]]
15	12, 14	a1i 28	. 2 ⊢ (R, T)⊧[(¬ S) = [S ⇒ ⊥]]
16	10, 15	mpbir 87	1 ⊢ (R, T)⊧(¬ S)

Syntax hints:  ∗hb 3  kc 5   = ke 7  [kbr 9  kct 10  ⊧wffMMJ2 11  wffMMJ2t 12  ⊥tfal 118  ¬ tne 120   ⇒ tim 121

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:  con3d  162  exnal1  187