Description: Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in Megill p. 445 (p. 12 of the preprint).
The original version of this axiom was ax-c11 and was replaced with this shorter ax-c11n ("n" for "new") in May 2008. The old axiom is proved from this one as theorem axc11 . Conversely, this axiom is proved from ax-c11 as theorem axc11nfromc11 .
This axiom was proved redundant in July 2015. See theorem axc11n .
This axiom is obsolete and should no longer be used. It is proved above as theorem axc11n . (Contributed by NM, 16-May-2008) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-c11n | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑦 = 𝑥 ) |
| 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 | 4 0 | wal | ⊢ ∀ 𝑥 𝑥 = 𝑦 |
| 6 | 3 1 | wceq | ⊢ 𝑦 = 𝑥 |
| 7 | 6 2 | wal | ⊢ ∀ 𝑦 𝑦 = 𝑥 |
| 8 | 5 7 | wi | ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑦 𝑦 = 𝑥 ) |