Description: Double restricted existential uniqueness, analogous to 2eu2ex . (Contributed by Alexander van der Vekens, 25-Jun-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | 2reu2rex | ⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reurex | ⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ) | |
2 | reurex | ⊢ ( ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑦 ∈ 𝐵 𝜑 ) | |
3 | 2 | reximi | ⊢ ( ∃ 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) |
4 | 1 3 | syl | ⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜑 ) |