Description: Universal quantification restricted to a Cartesian product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004) (Revised by Mario Carneiro, 29-Dec-2014)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ralxp.1 | ⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | ralxp | ⊢ ( ∀ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐵 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralxp.1 | ⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | iunxpconst | ⊢ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) = ( 𝐴 × 𝐵 ) | |
| 3 | 2 | raleqi | ⊢ ( ∀ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) 𝜑 ↔ ∀ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝜑 ) |
| 4 | 1 | raliunxp | ⊢ ( ∀ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐵 𝜓 ) |
| 5 | 3 4 | bitr3i | ⊢ ( ∀ 𝑥 ∈ ( 𝐴 × 𝐵 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐵 𝜓 ) |