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Mirrors > Home > HOLE Home > Th. List > exnal | GIF version |
Description: Theorem 19.14 of [Margaris] p. 90. (Contributed by Mario Carneiro, 10-Oct-2014.) |
Ref | Expression |
---|---|
exmid.1 | ⊢ A:∗ |
Ref | Expression |
---|---|
exnal | ⊢ ⊤⊧[(∃λx:α (¬ A)) = (¬ (∀λx:α A))] |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wnot 138 | . . 3 ⊢ ¬ :(∗ → ∗) | |
2 | wex 139 | . . . . 5 ⊢ ∃:((α → ∗) → ∗) | |
3 | exmid.1 | . . . . . . 7 ⊢ A:∗ | |
4 | 1, 3 | wc 50 | . . . . . 6 ⊢ (¬ A):∗ |
5 | 4 | wl 66 | . . . . 5 ⊢ λx:α (¬ A):(α → ∗) |
6 | 2, 5 | wc 50 | . . . 4 ⊢ (∃λx:α (¬ A)):∗ |
7 | 1, 6 | wc 50 | . . 3 ⊢ (¬ (∃λx:α (¬ A))):∗ |
8 | 1, 7 | wc 50 | . 2 ⊢ (¬ (¬ (∃λx:α (¬ A)))):∗ |
9 | wal 134 | . . . . 5 ⊢ ∀:((α → ∗) → ∗) | |
10 | 1, 4 | wc 50 | . . . . . 6 ⊢ (¬ (¬ A)):∗ |
11 | 10 | wl 66 | . . . . 5 ⊢ λx:α (¬ (¬ A)):(α → ∗) |
12 | 9, 11 | wc 50 | . . . 4 ⊢ (∀λx:α (¬ (¬ A))):∗ |
13 | 4 | alnex 186 | . . . 4 ⊢ ⊤⊧[(∀λx:α (¬ (¬ A))) = (¬ (∃λx:α (¬ A)))] |
14 | 12, 13 | eqcomi 79 | . . 3 ⊢ ⊤⊧[(¬ (∃λx:α (¬ A))) = (∀λx:α (¬ (¬ A)))] |
15 | 1, 7, 14 | ceq2 90 | . 2 ⊢ ⊤⊧[(¬ (¬ (∃λx:α (¬ A)))) = (¬ (∀λx:α (¬ (¬ A))))] |
16 | 6 | notnot 200 | . 2 ⊢ ⊤⊧[(∃λx:α (¬ A)) = (¬ (¬ (∃λx:α (¬ A))))] |
17 | 3 | wl 66 | . . . 4 ⊢ λx:α A:(α → ∗) |
18 | 9, 17 | wc 50 | . . 3 ⊢ (∀λx:α A):∗ |
19 | 3 | notnot 200 | . . . . 5 ⊢ ⊤⊧[A = (¬ (¬ A))] |
20 | 3, 19 | leq 91 | . . . 4 ⊢ ⊤⊧[λx:α A = λx:α (¬ (¬ A))] |
21 | 9, 17, 20 | ceq2 90 | . . 3 ⊢ ⊤⊧[(∀λx:α A) = (∀λx:α (¬ (¬ A)))] |
22 | 1, 18, 21 | ceq2 90 | . 2 ⊢ ⊤⊧[(¬ (∀λx:α A)) = (¬ (∀λx:α (¬ (¬ A))))] |
23 | 8, 15, 16, 22 | 3eqtr4i 96 | 1 ⊢ ⊤⊧[(∃λx:α (¬ A)) = (¬ (∀λx:α A))] |
Colors of variables: type var term |
Syntax hints: → ht 2 ∗hb 3 kc 5 λkl 6 = ke 7 ⊤kt 8 [kbr 9 ⊧wffMMJ2 11 wffMMJ2t 12 ¬ tne 120 ∀tal 122 ∃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 ax-eta 177 ax-wat 192 ax-ac 196 |
This theorem depends on definitions: df-ov 73 df-al 126 df-fal 127 df-an 128 df-im 129 df-not 130 df-ex 131 df-or 132 |
This theorem is referenced by: (None) |
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