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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 1601 ("o" for "old") and was replaced with this shorter ax-10 1393 in May 2008. The old axiom is proved from this one as theorem ax10o 1600. Conversely, this axiom is proved from ax-10o 1601 as theorem ax10 1602. (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 1389 | . . 3 wff x = y |
4 | 3, 1 | wal 1240 | . 2 wff ∀x x = y |
5 | 2, 1 | weq 1389 | . . 3 wff y = x |
6 | 5, 2 | wal 1240 | . 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 1405 ax10o 1600 naecoms 1609 oprabidlem 5479 |
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