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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 1458 ("o" for "old") and was replaced with this shorter ax10 1302 in May 2008. The old axiom is proved from this one as theorem ax10o 1457. Conversely, this axiom is proved from ax10o 1458 as theorem ax10 1459. (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 1298  . . 3 wff x = y 
4  3, 1  wal 1240  . 2 wff ∀x x = y 
5  2, 1  weq 1298  . . 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 1312 ax10o 1457 
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