Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfexd2 | Structured version Visualization version GIF version |
Description: Variation on nfexd 2348 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. Usage of this theorem is discouraged because it depends on ax-13 2390. Check out nfexd 2348 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) (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 1781 | . 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓) | |
2 | nfald2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfald2.2 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
4 | 3 | nfnd 1858 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 ¬ 𝜓) |
5 | 2, 4 | nfald2 2467 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ¬ 𝜓) |
6 | 5 | nfnd 1858 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓) |
7 | 1, 6 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1535 ∃wex 1780 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 |
This theorem is referenced by: nfeud2 2676 |
Copyright terms: Public domain | W3C validator |