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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
StepHypRef Expression
1 con2d.1 . . . . 5 |- T:*
2 wfal 125 . . . . 5 |- F.:*
3 con2d.2 . . . . . 6 |- (R, S) |= (~ T)
43ax-cb1 29 . . . . . . 7 |- (R, S):*
51notval 135 . . . . . . 7 |- T. |= [(~ T) = [T ==> F.]]
64, 5a1i 28 . . . . . 6 |- (R, S) |= [(~ T) = [T ==> F.]]
73, 6mpbi 72 . . . . 5 |- (R, S) |= [T ==> F.]
81, 2, 7imp 147 . . . 4 |- ((R, S), T) |= F.
98an32s 55 . . 3 |- ((R, T), S) |= F.
109ex 148 . 2 |- (R, T) |= [S ==> F.]
114wctl 31 . . . 4 |- R:*
1211, 1wct 44 . . 3 |- (R, T):*
134wctr 32 . . . 4 |- S:*
1413notval 135 . . 3 |- T. |= [(~ S) = [S ==> F.]]
1512, 14a1i 28 . 2 |- (R, T) |= [(~ S) = [S ==> F.]]
1610, 15mpbir 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  F.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
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