Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | eliind.a | ⊢ ( 𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ) | |
eliind.k | ⊢ ( 𝜑 → 𝐾 ∈ 𝐵 ) | ||
eliind.d | ⊢ ( 𝑥 = 𝐾 → ( 𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷 ) ) | ||
Assertion | eliind | ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliind.a | ⊢ ( 𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ) | |
2 | eliind.k | ⊢ ( 𝜑 → 𝐾 ∈ 𝐵 ) | |
3 | eliind.d | ⊢ ( 𝑥 = 𝐾 → ( 𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷 ) ) | |
4 | eliin | ⊢ ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) ) | |
5 | 1 4 | syl | ⊢ ( 𝜑 → ( 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) ) |
6 | 1 5 | mpbid | ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ) |
7 | 3 6 2 | rspcdva | ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 ) |