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Mirrors > Home > MPE Home > Th. List > Mathboxes > xfree2 | Structured version Visualization version GIF version |
Description: A partial converse to 19.9t 2202. (Contributed by Stefan Allan, 21-Dec-2008.) |
Ref | Expression |
---|---|
xfree2 | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xfree 30525 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) | |
2 | eximal 1790 | . . 3 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | |
3 | 2 | albii 1827 | . 2 ⊢ (∀𝑥(∃𝑥𝜑 → 𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
4 | 1, 3 | bitri 278 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1541 ∃wex 1787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-or 848 df-ex 1788 df-nf 1792 |
This theorem is referenced by: (None) |
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