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Theorem alnex 174
Description: Theorem 19.7 of [Margaris] p. 89.
Hypothesis
Ref Expression
alnex1.1 |- A:*
Assertion
Ref Expression
alnex |- T. |= [(A.\x:al (~ A)) = (~ (E.\x:al A))]
Distinct variable group:   al,x
Proof of Theorem alnex
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 alnex1.1 . . . . . 6 |- A:*
2 wfal 125 . . . . . 6 |- F.:*
3 wnot 128 . . . . . . . . 9 |- ~ :(* -> *)
43, 1wc 45 . . . . . . . 8 |- (~ A):*
54ax4 140 . . . . . . 7 |- (A.\x:al (~ A)) |= (~ A)
65ax-cb1 29 . . . . . . . 8 |- (A.\x:al (~ A)):*
71notval 135 . . . . . . . 8 |- T. |= [(~ A) = [A ==> F.]]
86, 7a1i 28 . . . . . . 7 |- (A.\x:al (~ A)) |= [(~ A) = [A ==> F.]]
95, 8mpbi 72 . . . . . 6 |- (A.\x:al (~ A)) |= [A ==> F.]
101, 2, 9imp 147 . . . . 5 |- ((A.\x:al (~ A)), A) |= F.
11 wal 124 . . . . . 6 |- A.:((al -> *) -> *)
124wl 59 . . . . . 6 |- \x:al (~ A):(al -> *)
13 wv 58 . . . . . 6 |- y:al:al
1411, 13ax-17 95 . . . . . 6 |- T. |= [(\x:al A.y:al) = A.]
154, 13ax-hbl1 93 . . . . . 6 |- T. |= [(\x:al \x:al (~ A)y:al) = \x:al (~ A)]
1611, 12, 13, 14, 15hbc 100 . . . . 5 |- T. |= [(\x:al (A.\x:al (~ A))y:al) = (A.\x:al (~ A))]
172, 13ax-17 95 . . . . 5 |- T. |= [(\x:al F.y:al) = F.]
1810, 16, 17exlimd 171 . . . 4 |- ((A.\x:al (~ A)), (E.\x:al A)) |= F.
1918ex 148 . . 3 |- (A.\x:al (~ A)) |= [(E.\x:al A) ==> F.]
20 wex 129 . . . . . 6 |- E.:((al -> *) -> *)
211wl 59 . . . . . 6 |- \x:al A:(al -> *)
2220, 21wc 45 . . . . 5 |- (E.\x:al A):*
2322notval 135 . . . 4 |- T. |= [(~ (E.\x:al A)) = [(E.\x:al A) ==> F.]]
246, 23a1i 28 . . 3 |- (A.\x:al (~ A)) |= [(~ (E.\x:al A)) = [(E.\x:al A) ==> F.]]
2519, 24mpbir 77 . 2 |- (A.\x:al (~ A)) |= (~ (E.\x:al A))
26119.8a 160 . . . . . 6 |- A |= (E.\x:al A)
27 wtru 40 . . . . . 6 |- T.:*
2826, 27adantl 51 . . . . 5 |- (T., A) |= (E.\x:al A)
2928con3d 152 . . . 4 |- (T., (~ (E.\x:al A))) |= (~ A)
3029trul 37 . . 3 |- (~ (E.\x:al A)) |= (~ A)
313, 13ax-17 95 . . . 4 |- T. |= [(\x:al ~ y:al) = ~ ]
3220, 13ax-17 95 . . . . 5 |- T. |= [(\x:al E.y:al) = E.]
331, 13ax-hbl1 93 . . . . 5 |- T. |= [(\x:al \x:al Ay:al) = \x:al A]
3420, 21, 13, 32, 33hbc 100 . . . 4 |- T. |= [(\x:al (E.\x:al A)y:al) = (E.\x:al A)]
353, 22, 13, 31, 34hbc 100 . . 3 |- T. |= [(\x:al (~ (E.\x:al A))y:al) = (~ (E.\x:al A))]
3630, 35alrimi 170 . 2 |- (~ (E.\x:al A)) |= (A.\x:al (~ A))
3725, 36dedi 75 1 |- T. |= [(A.\x:al (~ A)) = (~ (E.\x:al A))]
Colors of variables: type var term
Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 108  ~ tne 110   ==> tim 111  A.tal 112  E.tex 113
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-eta 165
This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120  df-ex 121
This theorem is referenced by:  exnal1  175  exnal  188  ax9  199
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