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Mirrors > Home > MPE Home > Th. List > nfrd | Structured version Visualization version GIF version |
Description: Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.) |
Ref | Expression |
---|---|
nfrd.1 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfrd | ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfrd.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
2 | df-nf 1778 | . 2 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓)) | |
3 | 1, 2 | sylib 217 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃wex 1773 Ⅎwnf 1777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-nf 1778 |
This theorem is referenced by: 19.38a 1834 19.38b 1835 nfimd 1889 nf5r 2179 19.9d 2188 nfald 2313 exists2 2649 eusv2i 5383 bj-nfimt 36006 bj-nfald 36509 |
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