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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 2638 for a version without disjoint variable conditions but requiring ax-13 2381. (Contributed 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 2615 | . 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
2 | nfv 1906 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfmodv.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfmodv.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | nfvd 1907 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
6 | 4, 5 | nfimd 1886 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝑦 = 𝑧)) |
7 | 3, 6 | nfald 2338 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝜓 → 𝑦 = 𝑧)) |
8 | 2, 7 | nfexd 2339 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
9 | 1, 8 | nfxfrd 1845 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 ∃*wmo 2613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-or 842 df-ex 1772 df-nf 1776 df-mo 2615 |
This theorem is referenced by: nfmov 2637 nfeudw 2670 nfrmow 3373 nfdisjw 5034 |
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