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Description: Axiom of Quantifier
Introduction. One of the equality and substitution
axioms of predicate calculus with equality. Informally, it says that
whenever z is distinct
from x and y, and x = y is
true,
then x = y quantified with z is also true. In other words, z
is irrelevant to the truth of x
= y. Axiom scheme C9' in [Megill]
p. 448 (p. 16 of the preprint). It apparently does not otherwise appear
in the literature but is easily proved from textbook predicate calculus by
cases.
This axiom has been modified from the original ax12 1318 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31Jan2015.) 
Ref  Expression 

axi12  ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) 
Step  Hyp  Ref  Expression 

1  vz  . . . 4 set z  
2  vx  . . . 4 set x  
3  1, 2  weq 1309  . . 3 wff z = x 
4  3, 1  wal 1250  . 2 wff ∀z z = x 
5  vy  . . . . 5 set y  
6  1, 5  weq 1309  . . . 4 wff z = y 
7  6, 1  wal 1250  . . 3 wff ∀z z = y 
8  2, 5  weq 1309  . . . . 5 wff x = y 
9  8, 1  wal 1250  . . . . 5 wff ∀z x = y 
10  8, 9  wi 4  . . . 4 wff (x = y → ∀z x = y) 
11  10, 1  wal 1250  . . 3 wff ∀z(x = y → ∀z x = y) 
12  7, 11  wo 605  . 2 wff (∀z z = y ∨ ∀z(x = y → ∀z x = y)) 
13  4, 12  wo 605  1 wff (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) 
Colors of variables: wff set class 
This axiom is referenced by: ax12 1318 ax12or 1319 dveeq2 1554 dveeq2or 1555 dvelimALT 1728 dvelimfv 1729 
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