Mathbox for Stefan Allan |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > xfree | Structured version Visualization version GIF version |
Description: A partial converse to 19.9t 2200. (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 2286 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
2 | nf6 2287 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) | |
3 | 1, 2 | bitr3i 279 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 ∃wex 1776 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1777 df-nf 1781 |
This theorem is referenced by: xfree2 30216 |
Copyright terms: Public domain | W3C validator |