Description: Obsolete version of r19.29r as of 29-Jun-2023. (Contributed by NM, 31-Aug-1999) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | r19.29rOLD | ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r19.29 | ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜓 ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜑 ) ) | |
| 2 | ancom | ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜓 ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) ) | |
| 3 | ancom | ⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) ) | |
| 4 | 3 | rexbii | ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜑 ) ) |
| 5 | 1 2 4 | 3imtr4i | ⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) |