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| Mirrors > Home > MPE Home > Th. List > hbe1 | Structured version Visualization version GIF version | ||
| Description: The setvar 𝑥 is not free in ∃𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2155). (Contributed by NM, 24-Jan-1993.) | 
| Ref | Expression | 
|---|---|
| hbe1 | ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ex 1780 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
| 2 | hbn1 2142 | . 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑) | |
| 3 | 1, 2 | hbxfrbi 1825 | 1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 ∃wex 1779 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-10 2141 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 | 
| This theorem is referenced by: nfe1 2150 equs5eALT 2370 nfeqf2 2382 equs5e 2463 axie1 2702 bj-wnf2 36719 bj-nnfe1 36761 ac6s6 38179 exlimexi 44544 vk15.4j 44548 vk15.4jVD 44934 | 
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