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Mirrors > Home > MPE Home > Th. List > Mathboxes > exlimddvf | Structured version Visualization version GIF version |
Description: A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.) |
Ref | Expression |
---|---|
exlimddvf.1 | ⊢ (𝜑 → ∃𝑥𝜃) |
exlimddvf.2 | ⊢ Ⅎ𝑥𝜓 |
exlimddvf.3 | ⊢ ((𝜃 ∧ 𝜓) → 𝜒) |
exlimddvf.4 | ⊢ Ⅎ𝑥𝜒 |
Ref | Expression |
---|---|
exlimddvf | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimddvf.1 | . 2 ⊢ (𝜑 → ∃𝑥𝜃) | |
2 | exlimddvf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | exlimddvf.4 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
4 | exlimddvf.3 | . . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒) | |
5 | 4 | expcom 451 | . . 3 ⊢ (𝜓 → (𝜃 → 𝜒)) |
6 | 2, 3, 5 | exlimd 2087 | . 2 ⊢ (𝜓 → (∃𝑥𝜃 → 𝜒)) |
7 | 1, 6 | mpan9 486 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∃wex 1704 Ⅎwnf 1708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: exlimddvfi 33927 |
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