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Theorem alrimi 182
 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.)
Hypotheses
Ref Expression
alrimi.1 RA
alrimi.2 ⊤⊧[(λx:α Ry:α) = R]
Assertion
Ref Expression
alrimi R⊧(λx:α A)
Distinct variable groups:   y,A   y,R   x,y,α

Proof of Theorem alrimi
StepHypRef Expression
1 alrimi.1 . . . 4 RA
21ax-cb2 30 . . 3 A:∗
3 wtru 43 . . . 4 ⊤:∗
41eqtru 86 . . . 4 R⊧[⊤ = A]
53, 4eqcomi 79 . . 3 R⊧[A = ⊤]
6 alrimi.2 . . 3 ⊤⊧[(λx:α Ry:α) = R]
72, 5, 6leqf 181 . 2 R⊧[λx:α A = λx:α ⊤]
81ax-cb1 29 . . 3 R:∗
92wl 66 . . . 4 λx:α A:(α → ∗)
109alval 142 . . 3 ⊤⊧[(λx:α A) = [λx:α A = λx:α ⊤]]
118, 10a1i 28 . 2 R⊧[(λx:α A) = [λx:α A = λx:α ⊤]]
127, 11mpbir 87 1 R⊧(λx:α A)
 Colors of variables: type var term Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  ∀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  ax-eta 177 This theorem depends on definitions:  df-ov 73  df-al 126 This theorem is referenced by:  alimdv  184  alnex  186  isfree  188  ax5  207  ax7  209
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