Description: Version of equs45f with a disjoint variable condition, which does not require ax-13 . Note that the version of equs5 with a disjoint variable condition is actually sb56 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | bj-equs45fv.1 | ⊢ Ⅎ 𝑦 𝜑 | |
| Assertion | bj-equs45fv | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-equs45fv.1 | ⊢ Ⅎ 𝑦 𝜑 | |
| 2 | 1 | nf5ri | ⊢ ( 𝜑 → ∀ 𝑦 𝜑 ) |
| 3 | 2 | anim2i | ⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) ) |
| 4 | 3 | eximi | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) ) |
| 5 | equs5av | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) | |
| 6 | 4 5 | syl | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
| 7 | equs4v | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) | |
| 8 | 6 7 | impbii | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |