Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvrexfw when possible. (Contributed by FL, 27-Apr-2008) (Revised by Mario Carneiro, 9-Oct-2016) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cbvralf.1 | ⊢ Ⅎ 𝑥 𝐴 | |
| cbvralf.2 | ⊢ Ⅎ 𝑦 𝐴 | ||
| cbvralf.3 | ⊢ Ⅎ 𝑦 𝜑 | ||
| cbvralf.4 | ⊢ Ⅎ 𝑥 𝜓 | ||
| cbvralf.5 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | ||
| Assertion | cbvrexf | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvralf.1 | ⊢ Ⅎ 𝑥 𝐴 | |
| 2 | cbvralf.2 | ⊢ Ⅎ 𝑦 𝐴 | |
| 3 | cbvralf.3 | ⊢ Ⅎ 𝑦 𝜑 | |
| 4 | cbvralf.4 | ⊢ Ⅎ 𝑥 𝜓 | |
| 5 | cbvralf.5 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| 6 | 3 | nfn | ⊢ Ⅎ 𝑦 ¬ 𝜑 |
| 7 | 4 | nfn | ⊢ Ⅎ 𝑥 ¬ 𝜓 |
| 8 | 5 | notbid | ⊢ ( 𝑥 = 𝑦 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) ) |
| 9 | 1 2 6 7 8 | cbvralf | ⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 ) |
| 10 | 9 | notbii | ⊢ ( ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 ) |
| 11 | dfrex2 | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 ) | |
| 12 | dfrex2 | ⊢ ( ∃ 𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀ 𝑦 ∈ 𝐴 ¬ 𝜓 ) | |
| 13 | 10 11 12 | 3bitr4i | ⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 𝜓 ) |