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Theorem exlimdv 167
Description: Existential elimination. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypothesis
Ref Expression
exlimdv.1 |- (R, A) |= T
Assertion
Ref Expression
exlimdv |- (R, (E.\x:al A)) |= T
Distinct variable groups:   x,R   x,T   al,x

Proof of Theorem exlimdv
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exlimdv.1 . . . . 5 |- (R, A) |= T
21ax-cb1 29 . . . 4 |- (R, A):*
32wctr 34 . . 3 |- A:*
43wl 66 . 2 |- \x:al A:(al -> *)
5 wv 64 . . . 4 |- y:al:al
64, 5wc 50 . . 3 |- (\x:al Ay:al):*
71ax-cb2 30 . . 3 |- T:*
8 wim 137 . . . . 5 |- ==> :(* -> (* -> *))
98, 6, 7wov 72 . . . 4 |- [(\x:al Ay:al) ==> T]:*
101ex 158 . . . 4 |- R |= [A ==> T]
11 wv 64 . . . . 5 |- z:al:al
128, 11ax-17 105 . . . . 5 |- T. |= [(\x:al ==> z:al) = ==> ]
133, 11ax-hbl1 103 . . . . . 6 |- T. |= [(\x:al \x:al Az:al) = \x:al A]
145, 11ax-17 105 . . . . . 6 |- T. |= [(\x:al y:alz:al) = y:al]
154, 5, 11, 13, 14hbc 110 . . . . 5 |- T. |= [(\x:al (\x:al Ay:al)z:al) = (\x:al Ay:al)]
167, 11ax-17 105 . . . . 5 |- T. |= [(\x:al Tz:al) = T]
178, 6, 11, 7, 12, 15, 16hbov 111 . . . 4 |- T. |= [(\x:al [(\x:al Ay:al) ==> T]z:al) = [(\x:al Ay:al) ==> T]]
18 wv 64 . . . . . . . 8 |- x:al:al
1918, 5weqi 76 . . . . . . 7 |- [x:al = y:al]:*
204, 18wc 50 . . . . . . . 8 |- (\x:al Ax:al):*
213beta 92 . . . . . . . 8 |- T. |= [(\x:al Ax:al) = A]
2220, 21eqcomi 79 . . . . . . 7 |- T. |= [A = (\x:al Ax:al)]
2319, 22a1i 28 . . . . . 6 |- [x:al = y:al] |= [A = (\x:al Ax:al)]
2419id 25 . . . . . . 7 |- [x:al = y:al] |= [x:al = y:al]
254, 18, 24ceq2 90 . . . . . 6 |- [x:al = y:al] |= [(\x:al Ax:al) = (\x:al Ay:al)]
263, 23, 25eqtri 95 . . . . 5 |- [x:al = y:al] |= [A = (\x:al Ay:al)]
278, 3, 7, 26oveq1 99 . . . 4 |- [x:al = y:al] |= [[A ==> T] = [(\x:al Ay:al) ==> T]]
285, 9, 10, 17, 27insti 114 . . 3 |- R |= [(\x:al Ay:al) ==> T]
296, 7, 28imp 157 . 2 |- (R, (\x:al Ay:al)) |= T
304, 29exlimdv2 166 1 |- (R, (E.\x:al A)) |= T
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  kct 10   |= wffMMJ2 11   ==> tim 121  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
This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129  df-ex 131
This theorem is referenced by: (None)
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