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Theorem alrimiv 141
Description: If one can prove RA where R does not contain x, then A is true for all x.
Hypothesis
Ref Expression
alrimiv.1 RA
Assertion
Ref Expression
alrimiv R⊧(λx:α A)
Distinct variable groups:   x,R   α,x

Proof of Theorem alrimiv
StepHypRef Expression
1 alrimiv.1 . . . 4 RA
21ax-cb2 30 . . 3 A:∗
3 wtru 40 . . . 4 ⊤:∗
41eqtru 76 . . . 4 R⊧[⊤ = A]
53, 4eqcomi 70 . . 3 R⊧[A = ⊤]
62, 5leq 81 . 2 R⊧[λx:α A = λx:α ⊤]
71ax-cb1 29 . . 3 R:∗
82wl 59 . . . 4 λx:α A:(α → ∗)
98alval 132 . . 3 ⊤⊧[(λx:α A) = [λx:α A = λx:α ⊤]]
107, 9a1i 28 . 2 R⊧[(λx:α A) = [λx:α A = λx:α ⊤]]
116, 10mpbir 77 1 R⊧(λx:α A)
Colors of variables: type var term
Syntax hints:  hb 3  kc 5  λkl 6   = ke 7  kt 8  [kbr 9  wffMMJ2 11  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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