Axiom ax-11 1394

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 1763). 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 1705, ax11v2 1698 and ax-11o 1701. (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 setvar x
2		vy	. . 3 setvar y
3	1, 2	weq 1389	. 2 wff x = y
4		wph	. . . 4 wff φ
5	4, 2	wal 1240	. . 3 wff ∀yφ
6	3, 4	wi 4	. . . 4 wff (x = y → φ)
7	6, 1	wal 1240	. . 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  1600  equs5a  1672  sbcof2  1688  ax11o  1700  ax11v  1705