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Mirrors > Home > MPE Home > Th. List > nfmod2 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for "at most one." (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
nfmod2.1 | ⊢ Ⅎ𝑦𝜑 |
nfmod2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfmod2 | ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2475 | . 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) | |
2 | nfmod2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfmod2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
4 | 2, 3 | nfexd2 2332 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
5 | 2, 3 | nfeud2 2482 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
6 | 4, 5 | nfimd 1823 | . 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 → ∃!𝑦𝜓)) |
7 | 1, 6 | nfxfrd 1780 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 ∃!weu 2470 ∃*wmo 2471 |
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-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-eu 2474 df-mo 2475 |
This theorem is referenced by: nfmod 2485 nfrmod 3113 nfrmo 3115 nfdisj 4632 |
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