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Mirrors > Home > MPE Home > Th. List > nfexd | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜓, then it is not free in ∃𝑦𝜓. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
nfald.1 | ⊢ Ⅎ𝑦𝜑 |
nfald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfexd | ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1783 | . 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓) | |
2 | nfald.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfald.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 3 | nfnd 1862 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) |
5 | 2, 4 | nfald 2322 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ¬ 𝜓) |
6 | 5 | nfnd 1862 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓) |
7 | 1, 6 | nfxfrd 1857 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 ∃wex 1782 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-or 847 df-ex 1783 df-nf 1787 |
This theorem is referenced by: nfmod2 2553 nfmodv 2554 nfeudw 2586 nfeld 2915 nfopabd 5217 nfttrcld 9705 axrepndlem1 10587 axrepndlem2 10588 axunndlem1 10590 axunnd 10591 axpowndlem2 10593 axpowndlem3 10594 axpowndlem4 10595 axregndlem2 10598 axinfndlem1 10600 axinfnd 10601 axacndlem4 10605 axacndlem5 10606 axacnd 10607 19.9d2rf 31711 hbexg 43317 |
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