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Mirrors > Home > MPE Home > Th. List > axc7e | Structured version Visualization version GIF version |
Description: Abbreviated version of axc7 2327 using the existential quantifier. Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Jul-2022.) |
Ref | Expression |
---|---|
axc7e | ⊢ (∃𝑥∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbe1a 2139 | . 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
2 | 1 | 19.21bi 2178 | 1 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 |
This theorem is referenced by: bj-19.12 33987 bj-axc10 34002 |
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