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Theorem cla4v 142
Description: If A(x) is true for all x:al, then it is true for C = A(B).
Hypotheses
Ref Expression
cla4v.1 |- A:*
cla4v.2 |- B:al
cla4v.3 |- [x:al = B] |= [A = C]
Assertion
Ref Expression
cla4v |- (A.\x:al A) |= C
Distinct variable groups:   x,B   x,C   al,x
Proof of Theorem cla4v
StepHypRef Expression
1 cla4v.1 . . . 4 |- A:*
21wl 59 . . 3 |- \x:al A:(al -> *)
3 cla4v.2 . . 3 |- B:al
42, 3ax4g 139 . 2 |- (A.\x:al A) |= (\x:al AB)
54ax-cb1 29 . . 3 |- (A.\x:al A):*
6 cla4v.3 . . . 4 |- [x:al = B] |= [A = C]
71, 3, 6cl 106 . . 3 |- T. |= [(\x:al AB) = C]
85, 7a1i 28 . 2 |- (A.\x:al A) |= [(\x:al AB) = C]
94, 8mpbi 72 1 |- (A.\x:al A) |= C
Colors of variables: type var term
Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  A.tal 112
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-hbl1 93  ax-17 95  ax-inst 103
This theorem depends on definitions:  df-ov 65  df-al 116
This theorem is referenced by:  pm2.21  143  ecase  153  exlimdv2  156  ax4e  158  eta  166  exlimd  171  ac  184  ax10  200
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