Description: Special case of equsexv proved from core axioms, ax-10 (modal5), and hba1 (modal4). (Contributed by BJ, 29-Dec-2020) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | bj-equsexval.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ ∀ 𝑥 𝜓 ) ) | |
| Assertion | bj-equsexval | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 𝜓 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-equsexval.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ ∀ 𝑥 𝜓 ) ) | |
| 2 | 1 | pm5.32i | ⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ( 𝑥 = 𝑦 ∧ ∀ 𝑥 𝜓 ) ) |
| 3 | 2 | exbii | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑥 𝜓 ) ) |
| 4 | ax6ev | ⊢ ∃ 𝑥 𝑥 = 𝑦 | |
| 5 | bj-19.41al | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑥 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝑦 ∧ ∀ 𝑥 𝜓 ) ) | |
| 6 | 4 5 | mpbiran | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑥 𝜓 ) ↔ ∀ 𝑥 𝜓 ) |
| 7 | 3 6 | bitri | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∀ 𝑥 𝜓 ) |