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Axiom ax-11o 1940
Description: Axiom ax-11o 1940 ("o" for "old") was the original version of ax-11 1624, before it was discovered (in Jan. 2007) that the shorter ax-11 1624 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."

This axiom is redundant, as shown by theorem ax11o 1939.

Normally, ax11o 1939 should be used rather than ax-11o 1940, except by theorems specifically studying the latter's properties. (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 1620 . . . 4  wff  x  =  y
43, 1wal 1532 . . 3  wff  A. x  x  =  y
54wn 5 . 2  wff  -.  A. x  x  =  y
6 wph . . . 4  wff  ph
73, 6wi 6 . . . . 5  wff  ( x  =  y  ->  ph )
87, 1wal 1532 . . . 4  wff  A. x
( x  =  y  ->  ph )
96, 8wi 6 . . 3  wff  ( ph  ->  A. x ( x  =  y  ->  ph )
)
103, 9wi 6 . 2  wff  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
115, 10wi 6 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  1941  ax11b  1942  a12study  27821  a12studyALT  27822  a12study3  27824
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