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Theorem ax9 212
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. (Contributed by Mario Carneiro, 10-Oct-2014.)
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 64 . . . . . 6 |- x:al:al
2 ax9.1 . . . . . 6 |- A:al
31, 2weqi 76 . . . . 5 |- [x:al = A]:*
4319.8a 170 . . . 4 |- [x:al = A] |= (E.\x:al [x:al = A])
5 wex 139 . . . . 5 |- E.:((al -> *) -> *)
63wl 66 . . . . 5 |- \x:al [x:al = A]:(al -> *)
7 wv 64 . . . . 5 |- y:al:al
85, 7ax-17 105 . . . . 5 |- T. |= [(\x:al E.y:al) = E.]
93, 7ax-hbl1 103 . . . . 5 |- T. |= [(\x:al \x:al [x:al = A]y:al) = \x:al [x:al = A]]
105, 6, 7, 8, 9hbc 110 . . . 4 |- T. |= [(\x:al (E.\x:al [x:al = A])y:al) = (E.\x:al [x:al = A])]
11 wtru 43 . . . . 5 |- T.:*
1211, 7ax-17 105 . . . 4 |- T. |= [(\x:al T.y:al) = T.]
135, 6wc 50 . . . . 5 |- (E.\x:al [x:al = A]):*
143, 13eqid 83 . . . 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 86 . . . . 5 |- [x:al = A] |= [T. = [x:al = A]]
1711, 16eqcomi 79 . . . 4 |- [x:al = A] |= [[x:al = A] = T.]
184, 10, 12, 14, 17ax-inst 113 . . 3 |- T. |= (E.\x:al [x:al = A])
1913notnot1 160 . . 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 138 . . 3 |- ~ :(* -> *)
22 wal 134 . . . 4 |- A.:((al -> *) -> *)
2321, 3wc 50 . . . . 5 |- (~ [x:al = A]):*
2423wl 66 . . . 4 |- \x:al (~ [x:al = A]):(al -> *)
2522, 24wc 50 . . 3 |- (A.\x:al (~ [x:al = A])):*
263alnex 186 . . 3 |- T. |= [(A.\x:al (~ [x:al = A])) = (~ (E.\x:al [x:al = A]))]
2721, 25, 26ceq2 90 . 2 |- T. |= [(~ (A.\x:al (~ [x:al = A]))) = (~ (~ (E.\x:al [x:al = A])))]
2820, 27mpbir 87 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 120  A.tal 122  E.tex 123
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  ax-eta 177
This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130  df-ex 131
This theorem is referenced by: (None)
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