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Mirrors > Home > ILE Home > Th. List > 19.9 | GIF version |
Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
Ref | Expression |
---|---|
19.9.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.9 | ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfri 1519 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | 2 | 19.9h 1643 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 Ⅎwnf 1460 ∃wex 1492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 |
This theorem depends on definitions: df-bi 117 df-nf 1461 |
This theorem is referenced by: alexim 1645 19.19 1666 19.36-1 1673 19.44 1682 19.45 1683 19.41 1686 |
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