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Mirrors > Home > ILE Home > Th. List > ax-ie2 | GIF version |
Description: Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true". One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
ax-ie2 | ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wps | . . . 4 wff 𝜓 | |
2 | vx | . . . . 5 setvar 𝑥 | |
3 | 1, 2 | wal 1346 | . . . 4 wff ∀𝑥𝜓 |
4 | 1, 3 | wi 4 | . . 3 wff (𝜓 → ∀𝑥𝜓) |
5 | 4, 2 | wal 1346 | . 2 wff ∀𝑥(𝜓 → ∀𝑥𝜓) |
6 | wph | . . . . 5 wff 𝜑 | |
7 | 6, 1 | wi 4 | . . . 4 wff (𝜑 → 𝜓) |
8 | 7, 2 | wal 1346 | . . 3 wff ∀𝑥(𝜑 → 𝜓) |
9 | 6, 2 | wex 1485 | . . . 4 wff ∃𝑥𝜑 |
10 | 9, 1 | wi 4 | . . 3 wff (∃𝑥𝜑 → 𝜓) |
11 | 8, 10 | wb 104 | . 2 wff (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) |
12 | 5, 11 | wi 4 | 1 wff (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) |
Colors of variables: wff set class |
This axiom is referenced by: 19.23ht 1490 bj-ex 13762 |
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