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Theorem eximdv 185
 Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypothesis
Ref Expression
alimdv.1 (R, A)⊧B
Assertion
Ref Expression
eximdv (R, (λx:α A))⊧(λx:α B)
Distinct variable groups:   x,R   α,x

Proof of Theorem eximdv
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 alimdv.1 . . 3 (R, A)⊧B
21ax-cb2 30 . . . 4 B:∗
3219.8a 170 . . 3 B⊧(λx:α B)
41, 3syl 16 . 2 (R, A)⊧(λx:α B)
51ax-cb1 29 . . . 4 (R, A):∗
65wctl 33 . . 3 R:∗
7 wv 64 . . 3 y:α:α
86, 7ax-17 105 . 2 ⊤⊧[(λx:α Ry:α) = R]
9 wex 139 . . 3 :((α → ∗) → ∗)
102wl 66 . . 3 λx:α B:(α → ∗)
119, 7ax-17 105 . . 3 ⊤⊧[(λx:α y:α) = ]
122, 7ax-hbl1 103 . . 3 ⊤⊧[(λx:α λx:α By:α) = λx:α B]
139, 10, 7, 11, 12hbc 110 . 2 ⊤⊧[(λx:α (λx:α B)y:α) = (λx:α B)]
144, 8, 13exlimd 183 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  ∃tex 123 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-distrl 70  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  df-an 128  df-im 129  df-ex 131 This theorem is referenced by: (None)
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