Description: Two ways of expressing substitution when y is not free in ph . The implication "to the left" is equs4 and does not require the nonfreeness hypothesis. Theorem sbalex replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 replaces it with a distinctor antecedent. (Contributed by NM, 25-Apr-2008) (Revised by Mario Carneiro, 4-Oct-2016) Usage of this theorem is discouraged because it depends on ax-13 . Use sbalex instead. (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | equs45f.1 | ⊢ Ⅎ 𝑦 𝜑 | |
| Assertion | equs45f | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equs45f.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 | equs5a | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) | |
| 6 | 4 5 | syl | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |
| 7 | equs4 | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) | |
| 8 | 6 7 | impbii | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) |