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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 2156). (Contributed by NM, 24-Jan-1993.) |
Ref | Expression |
---|---|
hbe1 | ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1787 | . 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
2 | hbn1 2142 | . 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑) | |
3 | 1, 2 | hbxfrbi 1831 | 1 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-10 2141 |
This theorem depends on definitions: df-bi 206 df-ex 1787 |
This theorem is referenced by: nfe1 2151 equs5eALT 2367 nfeqf2 2379 equs5e 2460 axie1 2705 bj-wnf2 34894 bj-nnfe1 34936 ac6s6 36324 exlimexi 42112 vk15.4j 42116 vk15.4jVD 42502 |
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