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Mirrors > Home > MPE Home > Th. List > nfmodv | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod 2620 for a version without disjoint variable conditions but requiring ax-13 2379. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by BJ, 28-Jan-2023.) |
Ref | Expression |
---|---|
nfmodv.1 | ⊢ Ⅎ𝑦𝜑 |
nfmodv.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfmodv | ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2598 | . 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfmodv.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfmodv.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | nfvd 1916 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
6 | 4, 5 | nfimd 1895 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧)) |
7 | 3, 6 | nfald 2336 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧)) |
8 | 2, 7 | nfexd 2337 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
9 | 1, 8 | nfxfrd 1855 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀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 |
This theorem depends on definitions: df-bi 210 df-or 845 df-ex 1782 df-nf 1786 df-mo 2598 |
This theorem is referenced by: nfmov 2619 nfeudw 2652 nfrmow 3328 nfdisjw 5007 |
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