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Mirrors > Home > NFE Home > Th. List > hbe1 | GIF version |
Description: x is not free in ∃xφ. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
hbe1 | ⊢ (∃xφ → ∀x∃xφ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1542 | . 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ) | |
2 | hbn1 1730 | . 2 ⊢ (¬ ∀x ¬ φ → ∀x ¬ ∀x ¬ φ) | |
3 | 1, 2 | hbxfrbi 1568 | 1 ⊢ (∃xφ → ∀x∃xφ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 ∃wex 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-6 1729 |
This theorem depends on definitions: df-bi 177 df-ex 1542 |
This theorem is referenced by: nfe1 1732 hba1 1786 19.23hOLD 1820 ax12olem5 1931 ax10lem2 1937 axie1 2328 hboprab1 5559 hboprab2 5560 hboprab3 5561 |
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