Axiom ax-11 1003

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 1305). 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 was ax-11o 1255 ("o" for "old") and was replaced with this shorter ax-11 1003 in Jan. 2007. The old axiom is proved from this one as theorem ax11o 1254. Conversely, this axiom is proved from ax-11o 1255 as theorem ax11 1256.

Juha Arpiainen proved the independence of this axiom (in the form of the older axiom ax-11o 1255) 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 1255 (from which the ax-11 1003 instance follows by theorem ax11 1256.) The proof is by induction on formula length, using ax11eq 1402 and ax11el 1403 for the basis steps and ax11indn 1405, ax11indi 1406, and ax11inda 1410 for the induction steps.

See also ax11v 1303 and ax11v2 1252 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 991	. . 3 class x
3		vy	. . . 4 set y
4	3	cv 991	. . 3 class y
5	2, 4	wceq 992	. 2 wff x = y
6		wph	. . . 4 wff φ
7	6, 3	wal 990	. . 3 wff ∀yφ
8	5, 6	wi 3	. . . 4 wff (x = y → φ)
9	8, 1	wal 990	. . 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 1008  ax10o 1176  equs5a 1234  equs5e 1235  ax11o 1254

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