Description: Theorem 19.41 of Margaris p. 90. See 19.41v for a version requiring fewer axioms. (Contributed by NM, 14-May-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 12-Jan-2018)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | 19.41.1 | ⊢ Ⅎ 𝑥 𝜓 | |
| Assertion | 19.41 | ⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.41.1 | ⊢ Ⅎ 𝑥 𝜓 | |
| 2 | 19.40 | ⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) ) | |
| 3 | 1 | 19.9 | ⊢ ( ∃ 𝑥 𝜓 ↔ 𝜓 ) |
| 4 | 3 | anbi2i | ⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ 𝜓 ) ) |
| 5 | 2 4 | sylib | ⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) → ( ∃ 𝑥 𝜑 ∧ 𝜓 ) ) |
| 6 | pm3.21 | ⊢ ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) | |
| 7 | 1 6 | eximd | ⊢ ( 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) |
| 8 | 7 | impcom | ⊢ ( ( ∃ 𝑥 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) |
| 9 | 5 8 | impbii | ⊢ ( ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ 𝜓 ) ) |