Description: Axiom of Left Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of an arbitrary binary predicate e. , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of Tarski, p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in Megill p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be.
We prove in ax8 that this axiom can be recovered from its weakened version ax8v where x and y are assumed to be disjoint variables. In particular, the only theorem referencing ax-8 should be ax8v . See the comment of ax8v for more details on these matters. (Contributed by NM, 30-Jun-1993) (Revised by BJ, 7-Dec-2020) Use ax8 instead. (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-8 | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vx | ⊢ 𝑥 | |
| 1 | 0 | cv | ⊢ 𝑥 |
| 2 | vy | ⊢ 𝑦 | |
| 3 | 2 | cv | ⊢ 𝑦 |
| 4 | 1 3 | wceq | ⊢ 𝑥 = 𝑦 |
| 5 | vz | ⊢ 𝑧 | |
| 6 | 5 | cv | ⊢ 𝑧 |
| 7 | 1 6 | wcel | ⊢ 𝑥 ∈ 𝑧 |
| 8 | 3 6 | wcel | ⊢ 𝑦 ∈ 𝑧 |
| 9 | 7 8 | wi | ⊢ ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) |
| 10 | 4 9 | wi | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) |