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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfexd | Structured version Visualization version GIF version |
Description: Variant of nfexd 2347. (Contributed by BJ, 25-Dec-2023.) |
Ref | Expression |
---|---|
bj-nfald.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
bj-nfald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
bj-nfexd | ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1780 | . 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓) | |
2 | bj-nfald.1 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
3 | bj-nfald.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 3 | nfnd 1857 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) |
5 | 2, 4 | bj-nfald 34451 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ¬ 𝜓) |
6 | 5 | nfnd 1857 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓) |
7 | 1, 6 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1534 ∃wex 1779 Ⅎwnf 1783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: copsex2d 34453 |
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