Description: Weak form of the LHS of bj-subst proved from the core axiom schemes. Compare ax12w . (Contributed by BJ, 26-May-2024) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | bj-substw.is | ⊢ ( 𝑥 = 𝑡 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | bj-substw | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-substw.is | ⊢ ( 𝑥 = 𝑡 → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | 1 | pm5.32i | ⊢ ( ( 𝑥 = 𝑡 ∧ 𝜑 ) ↔ ( 𝑥 = 𝑡 ∧ 𝜓 ) ) |
| 3 | 2 | exbii | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) ↔ ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜓 ) ) |
| 4 | 19.41v | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝑡 ∧ 𝜓 ) ) | |
| 5 | 3 4 | bitri | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) ↔ ( ∃ 𝑥 𝑥 = 𝑡 ∧ 𝜓 ) ) |
| 6 | 1 | biimprcd | ⊢ ( 𝜓 → ( 𝑥 = 𝑡 → 𝜑 ) ) |
| 7 | 6 | alrimiv | ⊢ ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |
| 8 | 5 7 | simplbiim | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |