Axiom ax-16 1828

Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1540 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory, but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-17 1540; see Theorem ax16 1827.

This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1827. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-16	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Detailed syntax breakdown of Axiom ax-16

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2		vy	. . . 4 setvar 𝑦
3	1, 2	weq 1517	. . 3 wff 𝑥 = 𝑦
4	3, 1	wal 1362	. 2 wff ∀𝑥 𝑥 = 𝑦
5		wph	. . 3 wff 𝜑
6	5, 1	wal 1362	. . 3 wff ∀𝑥𝜑
7	5, 6	wi 4	. 2 wff (𝜑 → ∀𝑥𝜑)
8	4, 7	wi 4	1 wff (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

This axiom is referenced by: (None)