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Mirrors > Home > MPE Home > Th. List > nfnf | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in Ⅎ𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
Ref | Expression |
---|---|
nfnf.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfnf | ⊢ Ⅎ𝑥Ⅎ𝑦𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1710 | . 2 ⊢ (Ⅎ𝑦𝜑 ↔ (∃𝑦𝜑 → ∀𝑦𝜑)) | |
2 | nfnf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | nfex 2154 | . . 3 ⊢ Ⅎ𝑥∃𝑦𝜑 |
4 | 2 | nfal 2153 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜑 |
5 | 3, 4 | nfim 1825 | . 2 ⊢ Ⅎ𝑥(∃𝑦𝜑 → ∀𝑦𝜑) |
6 | 1, 5 | nfxfr 1779 | 1 ⊢ Ⅎ𝑥Ⅎ𝑦𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 ∃wex 1704 Ⅎwnf 1708 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: nfnfc 2774 nfnfcALT 2775 bj-nfnfc 32853 bj-nfcf 32920 |
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