Description: Restricted quantifier version 19.41v . Version of r19.41 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 17-Dec-2003) Reduce dependencies on axioms. (Revised by BJ, 29-Mar-2020)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | r19.41v | ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-rex | ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜓 ) ) ) | |
| 2 | anass | ⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜓 ) ) ) | |
| 3 | 2 | exbii | ⊢ ( ∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜓 ) ) ) |
| 4 | 19.41v | ⊢ ( ∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) ) | |
| 5 | df-rex | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) | |
| 6 | 5 | bicomi | ⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑥 ∈ 𝐴 𝜑 ) |
| 7 | 4 6 | bianbi | ⊢ ( ∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ 𝜓 ) ) |
| 8 | 1 3 7 | 3bitr2i | ⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ 𝜓 ) ) |