Description: A restricted quantifier over an image set. Version of rexrnmpt with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 20-Aug-2015) (Revised by Gino Giotto, 26-Jan-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rexrnmptw.1 | ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) | |
| rexrnmptw.2 | ⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) ) | ||
| Assertion | rexrnmptw | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( ∃ 𝑦 ∈ ran 𝐹 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rexrnmptw.1 | ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) | |
| 2 | rexrnmptw.2 | ⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) ) | |
| 3 | 2 | notbid | ⊢ ( 𝑦 = 𝐵 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |
| 4 | 1 3 | ralrnmptw | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( ∀ 𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀ 𝑥 ∈ 𝐴 ¬ 𝜒 ) ) |
| 5 | 4 | notbid | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( ¬ ∀ 𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜒 ) ) |
| 6 | dfrex2 | ⊢ ( ∃ 𝑦 ∈ ran 𝐹 𝜓 ↔ ¬ ∀ 𝑦 ∈ ran 𝐹 ¬ 𝜓 ) | |
| 7 | dfrex2 | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜒 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜒 ) | |
| 8 | 5 6 7 | 3bitr4g | ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( ∃ 𝑦 ∈ ran 𝐹 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜒 ) ) |