Description: Transfer universal quantification from a variable x to another variable y contained in expression A . (Contributed by NM, 10-Jun-2005) (Revised by Mario Carneiro, 15-Aug-2014)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ralxfr.1 | ⊢ ( 𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵 ) | |
| ralxfr.2 | ⊢ ( 𝑥 ∈ 𝐵 → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) | ||
| ralxfr.3 | ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) | ||
| Assertion | ralxfr | ⊢ ( ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐶 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralxfr.1 | ⊢ ( 𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵 ) | |
| 2 | ralxfr.2 | ⊢ ( 𝑥 ∈ 𝐵 → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) | |
| 3 | ralxfr.3 | ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) | |
| 4 | 1 | adantl | ⊢ ( ( ⊤ ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 ) |
| 5 | 2 | adantl | ⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) |
| 6 | 3 | adantl | ⊢ ( ( ⊤ ∧ 𝑥 = 𝐴 ) → ( 𝜑 ↔ 𝜓 ) ) |
| 7 | 4 5 6 | ralxfrd | ⊢ ( ⊤ → ( ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐶 𝜓 ) ) |
| 8 | 7 | mptru | ⊢ ( ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐶 𝜓 ) |