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Theorem alrimiv 141
Description: If one can prove R |= A where R does not contain x, then A is true for all x.
Hypothesis
Ref Expression
alrimiv.1 |- R |= A
Assertion
Ref Expression
alrimiv |- R |= (A.\x:al A)
Distinct variable groups:   x,R   al,x
Proof of Theorem alrimiv
StepHypRef Expression
1 alrimiv.1 . . . 4 |- R |= A
21ax-cb2 30 . . 3 |- A:*
3 wtru 40 . . . 4 |- T.:*
41eqtru 76 . . . 4 |- R |= [T. = A]
53, 4eqcomi 70 . . 3 |- R |= [A = T.]
62, 5leq 81 . 2 |- R |= [\x:al A = \x:al T.]
71ax-cb1 29 . . 3 |- R:*
82wl 59 . . . 4 |- \x:al A:(al -> *)
98alval 132 . . 3 |- T. |= [(A.\x:al A) = [\x:al A = \x:al T.]]
107, 9a1i 28 . 2 |- R |= [(A.\x:al A) = [\x:al A = \x:al T.]]
116, 10mpbir 77 1 |- R |= (A.\x:al A)
Colors of variables: type var term
Syntax hints:  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  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:  exlimdv2  156  ax4e  158  exlimd  171  axgen  197  ax10  200  ax11  201  axrep  207  axpow  208  axun  209
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