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Axiom ax-11o 2175
Description: Axiom ax-11o 2175 ("o" for "old") was the original version of ax-11 1753, before it was discovered (in Jan. 2007) that the shorter ax-11 1753 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of " -.  A. x x  =  y  ->..." as informally meaning "if  x and  y are distinct variables then..." The antecedent becomes false if the same variable is substituted for  x and  y, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form  -.  A. x x  =  y a "distinctor."

Interestingly, if the wff expression substituted for  ph contains no wff variables, the resulting statement can be proved without invoking this axiom. This means that even though this axiom is metalogically independent from the others, it is not logically independent. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 2175 (from which the ax-11 1753 instance follows by theorem ax11 2189.) The proof is by induction on formula length, using ax11eq 2227 and ax11el 2228 for the basis steps and ax11indn 2229, ax11indi 2230, and ax11inda 2234 for the induction steps. (This paragraph is true provided we use ax-10o 2173 in place of ax-10 2174.)

This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 2028. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-11o  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )

Detailed syntax breakdown of Axiom ax-11o
StepHypRef Expression
1 vx . . . . 5  set  x
2 vy . . . . 5  set  y
31, 2weq 1650 . . . 4  wff  x  =  y
43, 1wal 1546 . . 3  wff  A. x  x  =  y
54wn 3 . 2  wff  -.  A. x  x  =  y
6 wph . . . 4  wff  ph
73, 6wi 4 . . . . 5  wff  ( x  =  y  ->  ph )
87, 1wal 1546 . . . 4  wff  A. x
( x  =  y  ->  ph )
96, 8wi 4 . . 3  wff  ( ph  ->  A. x ( x  =  y  ->  ph )
)
103, 9wi 4 . 2  wff  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
115, 10wi 4 1  wff  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
Colors of variables: wff set class
This axiom is referenced by:  ax11  2189
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