Description: Theorem 19.44 of Margaris p. 90. See 19.44v for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | 19.44.1 | ⊢ Ⅎ 𝑥 𝜓 | |
| Assertion | 19.44 | ⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.44.1 | ⊢ Ⅎ 𝑥 𝜓 | |
| 2 | 19.43 | ⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) ) | |
| 3 | 1 | 19.9 | ⊢ ( ∃ 𝑥 𝜓 ↔ 𝜓 ) |
| 4 | 3 | orbi2i | ⊢ ( ( ∃ 𝑥 𝜑 ∨ ∃ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ 𝜓 ) ) |
| 5 | 2 4 | bitri | ⊢ ( ∃ 𝑥 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ 𝜓 ) ) |