con2d - Higher-Order Logic Explorer

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Theorem con2d 151

Description: A contraposition deduction.

**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 125	. . . . 5 ⊢ ⊥:∗
3		con2d.2	. . . . . 6 ⊢ (R, S)⊧(¬ T)
4	3	ax-cb1 29	. . . . . . 7 ⊢ (R, S):∗
5	1	notval 135	. . . . . . 7 ⊢ ⊤⊧[(¬ T) = [T ⇒ ⊥]]
6	4, 5	a1i 28	. . . . . 6 ⊢ (R, S)⊧[(¬ T) = [T ⇒ ⊥]]
7	3, 6	mpbi 72	. . . . 5 ⊢ (R, S)⊧[T ⇒ ⊥]
8	1, 2, 7	imp 147	. . . 4 ⊢ ((R, S), T)⊧⊥
9	8	an32s 55	. . 3 ⊢ ((R, T), S)⊧⊥
10	9	ex 148	. 2 ⊢ (R, T)⊧[S ⇒ ⊥]
11	4	wctl 31	. . . 4 ⊢ R:∗
12	11, 1	wct 44	. . 3 ⊢ (R, T):∗
13	4	wctr 32	. . . 4 ⊢ S:∗
14	13	notval 135	. . 3 ⊢ ⊤⊧[(¬ S) = [S ⇒ ⊥]]
15	12, 14	a1i 28	. 2 ⊢ (R, T)⊧[(¬ S) = [S ⇒ ⊥]]
16	10, 15	mpbir 77	1 ⊢ (R, T)⊧(¬ S)

Colors of variables: type var term

Syntax hints: ∗hb 3 kc 5 = ke 7 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ⊥tfal 108 ¬ tne 110 ⇒ tim 111

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: con3d 152 exnal1 175