Axiom ax-11 964

Description: Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent ∀x(x = y → φ) is a way of expressing "y substituted for x in wff φ" (cf. sb6 1262). 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.

The original version of this axiom, that isn't otherwise used in our development, was ax-11o 1213 ("o" for "old"), which was replaced with this shorter ax-11 964 in Jan. 2007.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1213) from the others on 19-Jan-2006. See item 9a at http://us.metamath.org/award2003.html.

Interestingly, if the wff expression substituted for φ 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 1213 (from which the ax-11 964 instance follows by theorem ax11 1214.) The proof is by induction on formula length, using ax11eq 1356 and ax11el 1357 for the basis steps and ax11indn 1359, ax11indi 1360, and ax11inda 1364 for the induction steps.

See also ax11v 1260 and ax11v2 1210 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions.

Assertion

Ref	Expression
ax-11	⊢ (x = y → (∀yφ → ∀x(x = y → φ)))

Detailed syntax breakdown of Axiom ax-11

Step	Hyp	Ref	Expression
1		vx	. . . 4 set x
2	1	cv 952	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 952	. . 3 class y
5	2, 4	wceq 953	. 2 wff x = y
6		wph	. . . 4 wff φ
7	6, 3	wal 951	. . 3 wff ∀yφ
8	5, 6	wi 3	. . . 4 wff (x = y → φ)
9	8, 1	wal 951	. . 3 wff ∀x(x = y → φ)
10	7, 9	wi 3	. 2 wff (∀yφ → ∀x(x = y → φ))
11	5, 10	wi 3	1 wff (x = y → (∀yφ → ∀x(x = y → φ)))

This axiom is referenced by:  ax4 969  ax10o 1135  equs5a 1193  equs5e 1194  ax11o 1212

Copyright terms: Public domain