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Theorem alrimiv 151
Description: If one can prove R |= A where R does not contain x, then A is true for all x. (Contributed by Mario Carneiro, 9-Oct-2014.)
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 43 . . . 4 |- T.:*
41eqtru 86 . . . 4 |- R |= [T. = A]
53, 4eqcomi 79 . . 3 |- R |= [A = T.]
62, 5leq 91 . 2 |- R |= [\x:al A = \x:al T.]
71ax-cb1 29 . . 3 |- R:*
82wl 66 . . . 4 |- \x:al A:(al -> *)
98alval 142 . . 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 87 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 122
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-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
This theorem is referenced by:  exlimdv2  166  ax4e  168  exlimd  183  axgen  210  ax10  213  ax11  214  axrep  220  axpow  221  axun  222
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