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Theorem ax12 215
Description: Axiom of Quantifier Introduction. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). (Contributed by Mario Carneiro, 10-Oct-2014.)
Assertion
Ref Expression
ax12 |- T. |= [(~ (A.\z:al [z:al = x:al])) ==> [(~ (A.\z:al [z:al = y:al])) ==> [[x:al = y:al] ==> (A.\z:al [x:al = y:al])]]]
Distinct variable groups:   x,z   y,z   al,z

Proof of Theorem ax12
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wv 64 . . . . . . 7 |- x:al:al
2 wv 64 . . . . . . 7 |- y:al:al
31, 2weqi 76 . . . . . 6 |- [x:al = y:al]:*
4 wv 64 . . . . . . 7 |- p:al:al
53, 4ax-17 105 . . . . . 6 |- T. |= [(\z:al [x:al = y:al]p:al) = [x:al = y:al]]
63, 5isfree 188 . . . . 5 |- T. |= [[x:al = y:al] ==> (A.\z:al [x:al = y:al])]
7 wnot 138 . . . . . 6 |- ~ :(* -> *)
8 wal 134 . . . . . . 7 |- A.:((al -> *) -> *)
9 wv 64 . . . . . . . . 9 |- z:al:al
109, 2weqi 76 . . . . . . . 8 |- [z:al = y:al]:*
1110wl 66 . . . . . . 7 |- \z:al [z:al = y:al]:(al -> *)
128, 11wc 50 . . . . . 6 |- (A.\z:al [z:al = y:al]):*
137, 12wc 50 . . . . 5 |- (~ (A.\z:al [z:al = y:al])):*
146, 13adantr 55 . . . 4 |- (T., (~ (A.\z:al [z:al = y:al]))) |= [[x:al = y:al] ==> (A.\z:al [x:al = y:al])]
1514ex 158 . . 3 |- T. |= [(~ (A.\z:al [z:al = y:al])) ==> [[x:al = y:al] ==> (A.\z:al [x:al = y:al])]]
169, 1weqi 76 . . . . . 6 |- [z:al = x:al]:*
1716wl 66 . . . . 5 |- \z:al [z:al = x:al]:(al -> *)
188, 17wc 50 . . . 4 |- (A.\z:al [z:al = x:al]):*
197, 18wc 50 . . 3 |- (~ (A.\z:al [z:al = x:al])):*
2015, 19adantr 55 . 2 |- (T., (~ (A.\z:al [z:al = x:al]))) |= [(~ (A.\z:al [z:al = y:al])) ==> [[x:al = y:al] ==> (A.\z:al [x:al = y:al])]]
2120ex 158 1 |- T. |= [(~ (A.\z:al [z:al = x:al])) ==> [(~ (A.\z:al [z:al = y:al])) ==> [[x:al = y:al] ==> (A.\z:al [x:al = y:al])]]]
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  ~ tne 120   ==> tim 121  A.tal 122
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
This theorem is referenced by: (None)
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