Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016)
Ref | Expression | ||
---|---|---|---|
Hypothesis | cbvdisjv.1 | ⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) | |
Assertion | cbvdisjv | ⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvdisjv.1 | ⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) | |
2 | 1 | eleq2d | ⊢ ( 𝑥 = 𝑦 → ( 𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶 ) ) |
3 | 2 | cbvrmovw | ⊢ ( ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃* 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ) |
4 | 3 | albii | ⊢ ( ∀ 𝑧 ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀ 𝑧 ∃* 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ) |
5 | df-disj | ⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑧 ∃* 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ) | |
6 | df-disj | ⊢ ( Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀ 𝑧 ∃* 𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ) | |
7 | 4 5 6 | 3bitr4i | ⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶 ) |