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Mirrors > Home > HOLE Home > Th. List > eximdv | GIF version |
Description: Deduction from Theorem 19.22 of [Margaris] p. 90. |
Ref | Expression |
---|---|
alimdv.1 | ⊢ (R, A)⊧B |
Ref | Expression |
---|---|
eximdv | ⊢ (R, (∃λx:α A))⊧(∃λx:α B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alimdv.1 | . . 3 ⊢ (R, A)⊧B | |
2 | 1 | ax-cb2 30 | . . . 4 ⊢ B:∗ |
3 | 2 | 19.8a 160 | . . 3 ⊢ B⊧(∃λx:α B) |
4 | 1, 3 | syl 16 | . 2 ⊢ (R, A)⊧(∃λx:α B) |
5 | 1 | ax-cb1 29 | . . . 4 ⊢ (R, A):∗ |
6 | 5 | wctl 31 | . . 3 ⊢ R:∗ |
7 | wv 58 | . . 3 ⊢ y:α:α | |
8 | 6, 7 | ax-17 95 | . 2 ⊢ ⊤⊧[(λx:α Ry:α) = R] |
9 | wex 129 | . . 3 ⊢ ∃:((α → ∗) → ∗) | |
10 | 2 | wl 59 | . . 3 ⊢ λx:α B:(α → ∗) |
11 | 9, 7 | ax-17 95 | . . 3 ⊢ ⊤⊧[(λx:α ∃y:α) = ∃] |
12 | 2, 7 | ax-hbl1 93 | . . 3 ⊢ ⊤⊧[(λx:α λx:α By:α) = λx:α B] |
13 | 9, 10, 7, 11, 12 | hbc 100 | . 2 ⊢ ⊤⊧[(λx:α (∃λx:α B)y:α) = (∃λx:α B)] |
14 | 4, 8, 13 | exlimd 171 | 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 113 |
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 |
This theorem depends on definitions: df-ov 65 df-al 116 df-an 118 df-im 119 df-ex 121 |
This theorem is referenced by: (None) |
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