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Mirrors > Home > ILE Home > Th. List > ax-10 | GIF version |
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-10o 1658 ("o" for "old") and was replaced with this shorter ax-10 1448 in May 2008. The old axiom is proved from this one as theorem ax10o 1657. Conversely, this axiom is proved from ax-10o 1658 as theorem ax10 1659. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ax-10 | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . . 4 setvar 𝑥 | |
2 | vy | . . . 4 setvar 𝑦 | |
3 | 1, 2 | weq 1444 | . . 3 wff 𝑥 = 𝑦 |
4 | 3, 1 | wal 1294 | . 2 wff ∀𝑥 𝑥 = 𝑦 |
5 | 2, 1 | weq 1444 | . . 3 wff 𝑦 = 𝑥 |
6 | 5, 2 | wal 1294 | . 2 wff ∀𝑦 𝑦 = 𝑥 |
7 | 4, 6 | wi 4 | 1 wff (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff set class |
This axiom is referenced by: alequcom 1460 ax10o 1657 naecoms 1666 oprabidlem 5718 |
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