Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016) df-nf changed. (Revised by Wolf Lammen, 18-Sep-2021)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | albid.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| albid.2 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | ||
| Assertion | nfbidf | ⊢ ( 𝜑 → ( Ⅎ 𝑥 𝜓 ↔ Ⅎ 𝑥 𝜒 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | albid.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| 2 | albid.2 | ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) | |
| 3 | 1 2 | exbid | ⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑥 𝜒 ) ) |
| 4 | 1 2 | albid | ⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑥 𝜒 ) ) |
| 5 | 3 4 | imbi12d | ⊢ ( 𝜑 → ( ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝜒 → ∀ 𝑥 𝜒 ) ) ) |
| 6 | df-nf | ⊢ ( Ⅎ 𝑥 𝜓 ↔ ( ∃ 𝑥 𝜓 → ∀ 𝑥 𝜓 ) ) | |
| 7 | df-nf | ⊢ ( Ⅎ 𝑥 𝜒 ↔ ( ∃ 𝑥 𝜒 → ∀ 𝑥 𝜒 ) ) | |
| 8 | 5 6 7 | 3bitr4g | ⊢ ( 𝜑 → ( Ⅎ 𝑥 𝜓 ↔ Ⅎ 𝑥 𝜒 ) ) |