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Mirrors > Home > MPE Home > Th. List > nf5di | Structured version Visualization version GIF version |
Description: Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either Ⅎ𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.) |
Ref | Expression |
---|---|
nf5di.1 | ⊢ (𝜑 → Ⅎ𝑥𝜑) |
Ref | Expression |
---|---|
nf5di | ⊢ Ⅎ𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf5di.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜑) | |
2 | 1 | nf5rd 2054 | . . 3 ⊢ (𝜑 → (𝜑 → ∀𝑥𝜑)) |
3 | 2 | pm2.43i 50 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
4 | 3 | nf5i 2011 | 1 ⊢ Ⅎ𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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