Description: Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvral2vw when possible. (Contributed by NM, 10-Aug-2004) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cbvral2v.1 | ⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜒 ) ) | |
| cbvral2v.2 | ⊢ ( 𝑦 = 𝑤 → ( 𝜒 ↔ 𝜓 ) ) | ||
| Assertion | cbvral2v | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvral2v.1 | ⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜒 ) ) | |
| 2 | cbvral2v.2 | ⊢ ( 𝑦 = 𝑤 → ( 𝜒 ↔ 𝜓 ) ) | |
| 3 | 1 | ralbidv | ⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜒 ) ) |
| 4 | 3 | cbvralv | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ) |
| 5 | 2 | cbvralv | ⊢ ( ∀ 𝑦 ∈ 𝐵 𝜒 ↔ ∀ 𝑤 ∈ 𝐵 𝜓 ) |
| 6 | 5 | ralbii | ⊢ ( ∀ 𝑧 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜒 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 𝜓 ) |
| 7 | 4 6 | bitri | ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐵 𝜓 ) |