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Theorem ax9 199
Description: Axiom of Equality. Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.
Hypothesis
Ref Expression
ax9.1 |- A:al
Assertion
Ref Expression
ax9 |- T. |= (~ (A.\x:al (~ [x:al = A])))
Distinct variable group:   al,x
Proof of Theorem ax9
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 wv 58 . . . . . 6 |- x:al:al
2 ax9.1 . . . . . 6 |- A:al
31, 2weqi 68 . . . . 5 |- [x:al = A]:*
4319.8a 160 . . . 4 |- [x:al = A] |= (E.\x:al [x:al = A])
5 wex 129 . . . . 5 |- E.:((al -> *) -> *)
63wl 59 . . . . 5 |- \x:al [x:al = A]:(al -> *)
7 wv 58 . . . . 5 |- y:al:al
85, 7ax-17 95 . . . . 5 |- T. |= [(\x:al E.y:al) = E.]
93, 7ax-hbl1 93 . . . . 5 |- T. |= [(\x:al \x:al [x:al = A]y:al) = \x:al [x:al = A]]
105, 6, 7, 8, 9hbc 100 . . . 4 |- T. |= [(\x:al (E.\x:al [x:al = A])y:al) = (E.\x:al [x:al = A])]
11 wtru 40 . . . . 5 |- T.:*
1211, 7ax-17 95 . . . 4 |- T. |= [(\x:al T.y:al) = T.]
135, 6wc 45 . . . . 5 |- (E.\x:al [x:al = A]):*
143, 13eqid 73 . . . 4 |- [x:al = A] |= [(E.\x:al [x:al = A]) = (E.\x:al [x:al = A])]
153id 25 . . . . . 6 |- [x:al = A] |= [x:al = A]
1615eqtru 76 . . . . 5 |- [x:al = A] |= [T. = [x:al = A]]
1711, 16eqcomi 70 . . . 4 |- [x:al = A] |= [[x:al = A] = T.]
184, 10, 12, 14, 17ax-inst 103 . . 3 |- T. |= (E.\x:al [x:al = A])
1913notnot1 150 . . 3 |- (E.\x:al [x:al = A]) |= (~ (~ (E.\x:al [x:al = A])))
2018, 19syl 16 . 2 |- T. |= (~ (~ (E.\x:al [x:al = A])))
21 wnot 128 . . 3 |- ~ :(* -> *)
22 wal 124 . . . 4 |- A.:((al -> *) -> *)
2321, 3wc 45 . . . . 5 |- (~ [x:al = A]):*
2423wl 59 . . . 4 |- \x:al (~ [x:al = A]):(al -> *)
2522, 24wc 45 . . 3 |- (A.\x:al (~ [x:al = A])):*
263alnex 174 . . 3 |- T. |= [(A.\x:al (~ [x:al = A])) = (~ (E.\x:al [x:al = A]))]
2721, 25, 26ceq2 80 . 2 |- T. |= [(~ (A.\x:al (~ [x:al = A]))) = (~ (~ (E.\x:al [x:al = A])))]
2820, 27mpbir 77 1 |- T. |= (~ (A.\x:al (~ [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  ~ tne 110  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: (None)
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