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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
StepHypRef Expression
1 con2d.1 . . . . 5 T:∗
2 wfal 135 . . . . 5 ⊥:∗
3 con2d.2 . . . . . 6 (R, S)⊧(¬ T)
43ax-cb1 29 . . . . . . 7 (R, S):∗
51notval 145 . . . . . . 7 ⊤⊧[(¬ T) = [T ⇒ ⊥]]
64, 5a1i 28 . . . . . 6 (R, S)⊧[(¬ T) = [T ⇒ ⊥]]
73, 6mpbi 82 . . . . 5 (R, S)⊧[T ⇒ ⊥]
81, 2, 7imp 157 . . . 4 ((R, S), T)⊧⊥
98an32s 60 . . 3 ((R, T), S)⊧⊥
109ex 158 . 2 (R, T)⊧[S ⇒ ⊥]
114wctl 33 . . . 4 R:∗
1211, 1wct 48 . . 3 (R, T):∗
134wctr 34 . . . 4 S:∗
1413notval 145 . . 3 ⊤⊧[(¬ S) = [S ⇒ ⊥]]
1512, 14a1i 28 . 2 (R, T)⊧[(¬ S) = [S ⇒ ⊥]]
1610, 15mpbir 87 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 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
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