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 cbvex2v if possible. (Contributed by NM, 14-Sep-2003) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 16-Jun-2019) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cbval2.1 | ⊢ Ⅎ 𝑧 𝜑 | |
| cbval2.2 | ⊢ Ⅎ 𝑤 𝜑 | ||
| cbval2.3 | ⊢ Ⅎ 𝑥 𝜓 | ||
| cbval2.4 | ⊢ Ⅎ 𝑦 𝜓 | ||
| cbval2.5 | ⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) ) | ||
| Assertion | cbvex2 | ⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑧 ∃ 𝑤 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbval2.1 | ⊢ Ⅎ 𝑧 𝜑 | |
| 2 | cbval2.2 | ⊢ Ⅎ 𝑤 𝜑 | |
| 3 | cbval2.3 | ⊢ Ⅎ 𝑥 𝜓 | |
| 4 | cbval2.4 | ⊢ Ⅎ 𝑦 𝜓 | |
| 5 | cbval2.5 | ⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) ) | |
| 6 | 1 | nfn | ⊢ Ⅎ 𝑧 ¬ 𝜑 |
| 7 | 2 | nfn | ⊢ Ⅎ 𝑤 ¬ 𝜑 |
| 8 | 3 | nfn | ⊢ Ⅎ 𝑥 ¬ 𝜓 |
| 9 | 4 | nfn | ⊢ Ⅎ 𝑦 ¬ 𝜓 |
| 10 | 5 | notbid | ⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( ¬ 𝜑 ↔ ¬ 𝜓 ) ) |
| 11 | 6 7 8 9 10 | cbval2 | ⊢ ( ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 ¬ 𝜓 ) |
| 12 | 2nexaln | ⊢ ( ¬ ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ¬ 𝜑 ) | |
| 13 | 2nexaln | ⊢ ( ¬ ∃ 𝑧 ∃ 𝑤 𝜓 ↔ ∀ 𝑧 ∀ 𝑤 ¬ 𝜓 ) | |
| 14 | 11 12 13 | 3bitr4i | ⊢ ( ¬ ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ¬ ∃ 𝑧 ∃ 𝑤 𝜓 ) |
| 15 | 14 | con4bii | ⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑧 ∃ 𝑤 𝜓 ) |