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Mirrors > Home > MPE Home > Th. List > nf5-1 | Structured version Visualization version GIF version |
Description: One direction of nf5 2290 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.) |
Ref | Expression |
---|---|
nf5-1 | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exim 1834 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑)) | |
2 | hbe1a 2148 | . . 3 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | syl6 35 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑)) |
4 | 3 | nfd 1791 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∃wex 1780 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-10 2145 |
This theorem depends on definitions: df-bi 209 df-ex 1781 df-nf 1785 |
This theorem is referenced by: nf5i 2150 nf5dh 2151 nf5d 2292 hbnt 2302 19.9ht 2339 |
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