Axiom ax-11 1335

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 1682). 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.

Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1626, ax11v2 1619 and ax-11o 1622. (Contributed by NM, 5-Aug-1993.)

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	. . 3 set x
2		vy	. . 3 set y
3	1, 2	weq 1330	. 2 wff x = y
4		wph	. . . 4 wff φ
5	4, 2	wal 1271	. . 3 wff ∀yφ
6	3, 4	wi 4	. . . 4 wff (x = y → φ)
7	6, 1	wal 1271	. . 3 wff ∀x(x = y → φ)
8	5, 7	wi 4	. 2 wff (∀yφ → ∀x(x = y → φ))
9	3, 8	wi 4	1 wff (x = y → (∀yφ → ∀x(x = y → φ)))

This axiom is referenced by:  ax10o  1521  equs5a  1593  sbcof2  1609  ax11o  1621  ax11v  1626