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Theorem ax3 192
Description: Axiom Transp. Axiom A3 of [Margaris] p. 49.
Hypotheses
Ref Expression
ax3.1 |- R:*
ax3.2 |- S:*
Assertion
Ref Expression
ax3 |- T. |= [[(~ R) ==> (~ S)] ==> [S ==> R]]
Proof of Theorem ax3
StepHypRef Expression
1 ax3.1 . . . . 5 |- R:*
2 wnot 128 . . . . . 6 |- ~ :(* -> *)
32, 1wc 45 . . . . 5 |- (~ R):*
4 wim 127 . . . . . . . 8 |- ==> :(* -> (* -> *))
5 ax3.2 . . . . . . . . 9 |- S:*
62, 5wc 45 . . . . . . . 8 |- (~ S):*
74, 3, 6wov 64 . . . . . . 7 |- [(~ R) ==> (~ S)]:*
87, 5wct 44 . . . . . 6 |- ([(~ R) ==> (~ S)], S):*
91exmid 186 . . . . . 6 |- T. |= [R \/ (~ R)]
108, 9a1i 28 . . . . 5 |- ([(~ R) ==> (~ S)], S) |= [R \/ (~ R)]
1110ax-cb1 29 . . . . . 6 |- ([(~ R) ==> (~ S)], S):*
1211, 1simpr 23 . . . . 5 |- (([(~ R) ==> (~ S)], S), R) |= R
13 wfal 125 . . . . . . . 8 |- F.:*
147id 25 . . . . . . . . . 10 |- [(~ R) ==> (~ S)] |= [(~ R) ==> (~ S)]
153, 6, 14imp 147 . . . . . . . . 9 |- ([(~ R) ==> (~ S)], (~ R)) |= (~ S)
1615ax-cb1 29 . . . . . . . . . 10 |- ([(~ R) ==> (~ S)], (~ R)):*
175notval 135 . . . . . . . . . 10 |- T. |= [(~ S) = [S ==> F.]]
1816, 17a1i 28 . . . . . . . . 9 |- ([(~ R) ==> (~ S)], (~ R)) |= [(~ S) = [S ==> F.]]
1915, 18mpbi 72 . . . . . . . 8 |- ([(~ R) ==> (~ S)], (~ R)) |= [S ==> F.]
205, 13, 19imp 147 . . . . . . 7 |- (([(~ R) ==> (~ S)], (~ R)), S) |= F.
2120an32s 55 . . . . . 6 |- (([(~ R) ==> (~ S)], S), (~ R)) |= F.
221pm2.21 143 . . . . . 6 |- F. |= R
2321, 22syl 16 . . . . 5 |- (([(~ R) ==> (~ S)], S), (~ R)) |= R
241, 3, 1, 10, 12, 23ecase 153 . . . 4 |- ([(~ R) ==> (~ S)], S) |= R
2524ex 148 . . 3 |- [(~ R) ==> (~ S)] |= [S ==> R]
26 wtru 40 . . 3 |- T.:*
2725, 26adantl 51 . 2 |- (T., [(~ R) ==> (~ S)]) |= [S ==> R]
2827ex 148 1 |- T. |= [[(~ R) ==> (~ S)] ==> [S ==> R]]
Colors of variables: type var term
Syntax hints:  *hb 3  kc 5   = ke 7  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 108  ~ tne 110   ==> tim 111   \/ tor 114
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  ax-ac 183
This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120  df-or 122
This theorem is referenced by: (None)
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