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Mirrors > Home > MPE Home > Th. List > hbe1a | Structured version Visualization version GIF version |
Description: Dual statement of hbe1 2008. Modified version of axc7e 2118 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.) |
Ref | Expression |
---|---|
hbe1a | ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1696 | . 2 ⊢ (∃𝑥∀𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑) | |
2 | hbn1 2007 | . . 3 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | |
3 | 2 | con1i 143 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) |
4 | 1, 3 | sylbi 206 | 1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-10 2006 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: nf5-1 2010 axc7e 2118 wl-dveeq12 32490 |
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