Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-nf-nf2 | Structured version Visualization version GIF version |
Description: By ax-10 2006 the definition nf5 2102 is stricter than the df-nf 1701 style. (Contributed by Wolf Lammen, 14-Sep-2021.) |
Ref | Expression |
---|---|
wl-nf-nf2.1 | ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
Ref | Expression |
---|---|
wl-nf-nf2 | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exim 1751 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑)) | |
2 | df-ex 1696 | . . 3 ⊢ (∃𝑥∀𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑) | |
3 | wl-nf-nf2.1 | . . . 4 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | |
4 | 3 | con1i 143 | . . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) |
5 | 2, 4 | sylbi 206 | . 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) |
6 | 1, 5 | syl6 34 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃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-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: (None) |
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