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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 1576 ("o" for "old") and was replaced with this shorter ax10 1388 in May 2008. The old axiom is proved from this one as theorem ax10o 1575. Conversely, this axiom is proved from ax10o 1576 as theorem ax10 1577. 
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 1384  . . 3 wff x = y 
4  3, 1  wal 1335  . 2 wff ∀x x = y 
5  2, 1  weq 1384  . . 3 wff y = x 
6  5, 2  wal 1335  . 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 1399 a9wa9lem8 1413 a9wa9lem8OLD 1414 ax10o 1575 
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