Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 4-Oct-2016) (Revised by BJ, 17-Jun-2019) Avoid ax-10 . (Revised by Gino Giotto, 18-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | dral1v.1 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | drnf1v | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑥 𝜑 ↔ Ⅎ 𝑦 𝜓 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dral1v.1 | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | 1 | drex1v | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜓 ) ) |
| 3 | 1 | dral1v | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) ) |
| 4 | 2 3 | imbi12d | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ↔ ( ∃ 𝑦 𝜓 → ∀ 𝑦 𝜓 ) ) ) |
| 5 | df-nf | ⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ) | |
| 6 | df-nf | ⊢ ( Ⅎ 𝑦 𝜓 ↔ ( ∃ 𝑦 𝜓 → ∀ 𝑦 𝜓 ) ) | |
| 7 | 4 5 6 | 3bitr4g | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( Ⅎ 𝑥 𝜑 ↔ Ⅎ 𝑦 𝜓 ) ) |