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Theorem cla4ev 159
Description: Existential introduction.
Hypotheses
Ref Expression
cla4ev.1 |- A:*
cla4ev.2 |- B:al
cla4ev.3 |- [x:al = B] |= [A = C]
Assertion
Ref Expression
cla4ev |- C |= (E.\x:al A)
Distinct variable groups:   x,B   x,C   al,x

Proof of Theorem cla4ev
StepHypRef Expression
1 cla4ev.1 . . . . 5 |- A:*
2 cla4ev.3 . . . . 5 |- [x:al = B] |= [A = C]
31, 2eqtypi 69 . . . 4 |- C:*
43id 25 . . 3 |- C |= C
5 cla4ev.2 . . . . 5 |- B:al
61, 5, 2cl 106 . . . 4 |- T. |= [(\x:al AB) = C]
73, 6a1i 28 . . 3 |- C |= [(\x:al AB) = C]
84, 7mpbir 77 . 2 |- C |= (\x:al AB)
91wl 59 . . 3 |- \x:al A:(al -> *)
109, 5ax4e 158 . 2 |- (\x:al AB) |= (E.\x:al A)
118, 10syl 16 1 |- C |= (E.\x:al A)
Colors of variables: type var term
Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  E.tex 113
This theorem is referenced by:  axpow  208  axun  209
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
This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119  df-ex 121
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