Description: It is possible to remove any restriction on ph in ax12v . Same as Axiom C8 of Monk2 p. 105. Use ax12v instead when sufficient. (Contributed by NM, 5-Aug-1993) Remove dependencies on ax-10 and ax-13 . (Revised by Jim Kingdon, 15-Dec-2017) (Proof shortened by Wolf Lammen, 8-Dec-2019)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax12v2 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | equtrr | ⊢ ( 𝑦 = 𝑧 → ( 𝑥 = 𝑦 → 𝑥 = 𝑧 ) ) | |
| 2 | ax12v | ⊢ ( 𝑥 = 𝑧 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) ) ) | |
| 3 | 1 | imim1d | ⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 = 𝑧 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 4 | 3 | alimdv | ⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |
| 5 | 2 4 | syl9r | ⊢ ( 𝑦 = 𝑧 → ( 𝑥 = 𝑧 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) ) |
| 6 | 1 5 | syld | ⊢ ( 𝑦 = 𝑧 → ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) ) |
| 7 | ax6evr | ⊢ ∃ 𝑧 𝑦 = 𝑧 | |
| 8 | 6 7 | exlimiiv | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) ) |