Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever z is distinct from x and y , and x = y is true, then x = y quantified with z is also true. In other words, z is irrelevant to the truth of x = y . Axiom scheme C9' in Megill p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.
This axiom is obsolete and should no longer be used. It is proved above as theorem axc9 . (Contributed by NM, 10-Jan-1993) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-c9 | ⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vz | ⊢ 𝑧 | |
| 1 | 0 | cv | ⊢ 𝑧 |
| 2 | vx | ⊢ 𝑥 | |
| 3 | 2 | cv | ⊢ 𝑥 |
| 4 | 1 3 | wceq | ⊢ 𝑧 = 𝑥 |
| 5 | 4 0 | wal | ⊢ ∀ 𝑧 𝑧 = 𝑥 |
| 6 | 5 | wn | ⊢ ¬ ∀ 𝑧 𝑧 = 𝑥 |
| 7 | vy | ⊢ 𝑦 | |
| 8 | 7 | cv | ⊢ 𝑦 |
| 9 | 1 8 | wceq | ⊢ 𝑧 = 𝑦 |
| 10 | 9 0 | wal | ⊢ ∀ 𝑧 𝑧 = 𝑦 |
| 11 | 10 | wn | ⊢ ¬ ∀ 𝑧 𝑧 = 𝑦 |
| 12 | 3 8 | wceq | ⊢ 𝑥 = 𝑦 |
| 13 | 12 0 | wal | ⊢ ∀ 𝑧 𝑥 = 𝑦 |
| 14 | 12 13 | wi | ⊢ ( 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) |
| 15 | 11 14 | wi | ⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) ) |
| 16 | 6 15 | wi | ⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑥 → ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) ) ) |