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Mirrors > Home > MPE Home > Th. List > Mathboxes > xfree | Structured version Visualization version GIF version |
Description: A partial converse to 19.9t 2171. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
xfree | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nf5 2258 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
2 | nf6 2259 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) | |
3 | 1, 2 | bitr3i 278 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1523 ∃wex 1765 Ⅎwnf 1769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-10 2114 ax-12 2143 |
This theorem depends on definitions: df-bi 208 df-or 843 df-ex 1766 df-nf 1770 |
This theorem is referenced by: xfree2 29909 |
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