Description: Version of rexeqbidv with additional disjoint variable conditions, not requiring ax-8 nor df-clel . (Contributed by Wolf Lammen, 25-Sep-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | raleqbidvv.1 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
| raleqbidvv.2 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | ||
| Assertion | rexeqbidvv | ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raleqbidvv.1 | ⊢ ( 𝜑 → 𝐴 = 𝐵 ) | |
| 2 | raleqbidvv.2 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
| 3 | 2 | notbid | ⊢ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |
| 4 | 1 3 | raleqbidvv | ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝜒 ) ) |
| 5 | ralnex | ⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝜓 ) | |
| 6 | ralnex | ⊢ ( ∀ 𝑥 ∈ 𝐵 ¬ 𝜒 ↔ ¬ ∃ 𝑥 ∈ 𝐵 𝜒 ) | |
| 7 | 4 5 6 | 3bitr3g | ⊢ ( 𝜑 → ( ¬ ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∃ 𝑥 ∈ 𝐵 𝜒 ) ) |
| 8 | 7 | con4bid | ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐵 𝜒 ) ) |