Axiom ax-c15 2220

Description: Axiom ax-c15 2220 was the original version of ax-12 1854, before it was discovered (in Jan. 2007) that the shorter ax-12 1854 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.xx=->..." 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."

Interestingly, if the wff expression substituted for 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-c15 2220 (from which the ax-12 1854 instance follows by theorem ax12 2234.) The proof is by induction on formula length, using ax12eq 2271 and ax12el 2272 for the basis steps and ax12indn 2273, ax12indi 2274, and ax12inda 2278 for the induction steps. (This paragraph is true provided we use ax-c11 2218 in place of ax-c11n 2219.)

This axiom is obsolete and should no longer be used. It is proved above as theorem axc15 2085, which should be used instead. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c15

Detailed syntax breakdown of Axiom ax-c15

Step	Hyp	Ref	Expression
1		vx	. . . . 5
2		vy	. . . . 5
3	1, 2	weq 1733	. . . 4
4	3, 1	wal 1393	. . 3
5	4	wn 3	. 2
6		wph	. . . 4
7	3, 6	wi 4	. . . . 5
8	7, 1	wal 1393	. . . 4
9	6, 8	wi 4	. . 3
10	3, 9	wi 4	. 2
11	5, 10	wi 4	1

This axiom is referenced by:  ax12  2234