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Mirrors > Home > ILE Home > Th. List > 19.9h | 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.) |
Ref | Expression |
---|---|
19.9h.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
19.9h | ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9ht 1628 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | |
2 | 19.9h.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | mpg 1438 | . 2 ⊢ (∃𝑥𝜑 → 𝜑) |
4 | 19.8a 1577 | . 2 ⊢ (𝜑 → ∃𝑥𝜑) | |
5 | 3, 4 | impbii 125 | 1 ⊢ (∃𝑥𝜑 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 ∃wex 1479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.9 1631 excomim 1650 exdistrfor 1787 sbcof2 1797 ax11ev 1815 19.9v 1858 exists1 2109 |
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