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Theorem alimdv 172
Description: Deduction from Theorem 19.20 of [Margaris] p. 90.
Hypothesis
Ref Expression
alimdv.1 (R, A)⊧B
Assertion
Ref Expression
alimdv (R, (λx:α A))⊧(λx:α B)
Distinct variable groups:   x,R   α,x

Proof of Theorem alimdv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 alimdv.1 . . . . . . 7 (R, A)⊧B
21ax-cb1 29 . . . . . 6 (R, A):∗
32wctr 32 . . . . 5 A:∗
43ax4 140 . . . 4 (λx:α A)⊧A
52wctl 31 . . . 4 R:∗
64, 5adantl 51 . . 3 (R, (λx:α A))⊧A
76, 1syldan 34 . 2 (R, (λx:α A))⊧B
8 wv 58 . . 3 y:α:α
9 wal 124 . . . 4 :((α → ∗) → ∗)
103wl 59 . . . 4 λx:α A:(α → ∗)
119, 10wc 45 . . 3 (λx:α A):∗
125, 8ax-17 95 . . 3 ⊤⊧[(λx:α Ry:α) = R]
139, 8ax-17 95 . . . 4 ⊤⊧[(λx:α y:α) = ]
143, 8ax-hbl1 93 . . . 4 ⊤⊧[(λx:α λx:α Ay:α) = λx:α A]
159, 10, 8, 13, 14hbc 100 . . 3 ⊤⊧[(λx:α (λx:α A)y:α) = (λx:α A)]
165, 8, 11, 12, 15hbct 145 . 2 ⊤⊧[(λx:α (R, (λx:α A))y:α) = (R, (λx:α A))]
177, 16alrimi 170 1 (R, (λx:α A))⊧(λx:α B)
Colors of variables: type var term
Syntax hints:  tv 1  ht 2  hb 3  kc 5  λkl 6  kt 8  kct 10  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-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-eta 165
This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118
This theorem is referenced by:  exnal1  175
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