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Mirrors > Home > MPE Home > Th. List > nfmod2 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfmodv 2618 for a version replacing the distinctor with a disjoint variable condition, not requiring ax-13 2379. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid df-eu 2629. (Revised by BJ, 14-Oct-2022.) (New usage is discouraged.) |
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 2598 | . 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfmod2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfmod2.2 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
5 | nfeqf1 2386 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) | |
6 | 5 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 = 𝑧) |
7 | 4, 6 | nfimd 1895 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧)) |
8 | 3, 7 | nfald2 2456 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧)) |
9 | 2, 8 | nfexd 2337 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
10 | 1, 9 | nfxfrd 1855 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∀wal 1536 ∃wex 1781 Ⅎwnf 1785 ∃*wmo 2596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-mo 2598 |
This theorem is referenced by: nfmod 2620 nfeud2 2651 nfrmod 3326 nfrmo 3330 nfdisj 5008 |
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