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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvelimexcasei | Structured version Visualization version GIF version |
Description: Eliminate a disjoint variable condition from an existentially quantified statement using cases. Inference form of dvelimexcased 35045. See axnulg 35060 for an example of its use. (Contributed by BTernaryTau, 31-Jul-2025.) |
Ref | Expression |
---|---|
dvelimexcasei.1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑) |
dvelimexcasei.2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒) |
dvelimexcasei.3 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒))) |
dvelimexcasei.4 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒)) |
dvelimexcasei.5 | ⊢ ∃𝑧𝜑 |
dvelimexcasei.6 | ⊢ ∃𝑥𝜓 |
Ref | Expression |
---|---|
dvelimexcasei | ⊢ ∃𝑥𝜒 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1802 | . . 3 ⊢ Ⅎ𝑥⊤ | |
2 | nfvd 1914 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧⊤) | |
3 | dvelimexcasei.1 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜑) |
5 | dvelimexcasei.2 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒) | |
6 | 5 | adantl 481 | . . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜒) |
7 | dvelimexcasei.3 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒))) | |
8 | 7 | adantl 481 | . . 3 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜑 → 𝜒))) |
9 | dvelimexcasei.4 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒)) | |
10 | 9 | adantl 481 | . . 3 ⊢ ((⊤ ∧ ∀𝑥 𝑥 = 𝑦) → (𝜓 → 𝜒)) |
11 | dvelimexcasei.5 | . . . 4 ⊢ ∃𝑧𝜑 | |
12 | 11 | a1i 11 | . . 3 ⊢ (⊤ → ∃𝑧𝜑) |
13 | dvelimexcasei.6 | . . . 4 ⊢ ∃𝑥𝜓 | |
14 | 13 | a1i 11 | . . 3 ⊢ (⊤ → ∃𝑥𝜓) |
15 | 1, 2, 4, 6, 8, 10, 12, 14 | dvelimexcased 35045 | . 2 ⊢ (⊤ → ∃𝑥𝜒) |
16 | 15 | mptru 1544 | 1 ⊢ ∃𝑥𝜒 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 ⊤wtru 1538 ∃wex 1777 Ⅎwnf 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-10 2136 ax-11 2153 ax-12 2173 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-ex 1778 df-nf 1782 |
This theorem is referenced by: axsepg2 35050 axsepg2ALT 35051 axnulg 35060 |
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