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Axiom ax-11o 1654
Description: Axiom ax-11o 1654 ("o" for "old") was the original version of ax-11 1389, before it was discovered (in Jan. 2007) that the shorter ax-11 1389 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 " ..." as informally meaning "if and are distinct variables then..." The antecedent becomes false if the same variable is substituted for and , ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form a "distinctor."

This axiom is redundant, as shown by theorem ax11o 1653.

Normally, ax11o 1653 should be used rather than ax-11o 1654, except by theorems specifically studying the latter's properties.

Assertion
Ref Expression
ax-11o

Detailed syntax breakdown of Axiom ax-11o
StepHypRef Expression
1 vx . . . . 5
2 vy . . . . 5
31, 2weq 1384 . . . 4
43, 1wal 1335 . . 3
54wn 3 . 2
6 wph . . . 4
73, 6wi 4 . . . . 5
87, 1wal 1335 . . . 4
96, 8wi 4 . . 3
103, 9wi 4 . 2
115, 10wi 4 1
Colors of variables: wff set class
This axiom is referenced by:  ax11  1655  ax11b  1656  a12study  1825  a12studyALT  1826  a12study3  1828
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