Description: Transfer existential quantification from a variable x to another variable y contained in expression A . (Contributed by SN, 20-Jun-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rexxfr3d.s | ⊢ ( 𝑥 = 𝑋 → ( 𝜓 ↔ 𝜒 ) ) | |
| rexxfr3d.x | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 ) ) | ||
| rexxfr3d.a | ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) | ||
| Assertion | rexxfr3d | ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexxfr3d.s | ⊢ ( 𝑥 = 𝑋 → ( 𝜓 ↔ 𝜒 ) ) | |
| 2 | rexxfr3d.x | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = 𝑋 ) ) | |
| 3 | rexxfr3d.a | ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) | |
| 4 | 3 | adantr | ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑋 ∈ 𝑉 ) | 
| 5 | 1 | adantl | ⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝜓 ↔ 𝜒 ) ) | 
| 6 | 4 2 5 | rexxfr2d | ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑦 ∈ 𝐵 𝜒 ) ) |