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Mirrors > Home > ILE Home > Th. List > ax10  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 ax10o 1427 ("o" for "old") and was replaced with this shorter ax10 1284 in May 2008. The old axiom is proved from this one as theorem ax10o 1426. Conversely, this axiom is proved from ax10o 1427 as theorem ax10 1428. (Contributed by NM, 5Aug1993.) 
Ref  Expression 

ax10  ⊢ (∀x x = y → ∀y y = x) 
Step  Hyp  Ref  Expression 

1  vx  . . . 4 set x  
2  vy  . . . 4 set y  
3  1, 2  weq 1280  . . 3 wff x = y 
4  3, 1  wal 1231  . 2 wff ∀x x = y 
5  2, 1  weq 1280  . . 3 wff y = x 
6  5, 2  wal 1231  . 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 1294 ax10o 1426 ax10oe 1433 
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