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Mirrors > Home > MPE Home > Th. List > nfexd2 | Structured version Visualization version GIF version |
Description: Variation on nfexd 2318 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2367. Use nfexd 2318 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
nfald2.1 | ⊢ Ⅎ𝑦𝜑 |
nfald2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfexd2 | ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1775 | . 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓) | |
2 | nfald2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfald2.2 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
4 | 3 | nfnd 1854 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 ¬ 𝜓) |
5 | 2, 4 | nfald2 2440 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ¬ 𝜓) |
6 | 5 | nfnd 1854 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓) |
7 | 1, 6 | nfxfrd 1849 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1532 ∃wex 1774 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 |
This theorem is referenced by: nfeud2 2580 |
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