Axiom ax-c16 36643

Description: Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-5 1918 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 (see dtru 5263), but nonetheless it is technically necessary as you can see from its uses.

This axiom is redundant if we include ax-5 1918; see Theorem axc16 2258. Alternately, ax-5 1918 becomes logically redundant in the presence of this axiom, but without ax-5 1918 we lose the more powerful metalogic that results from being able to express the concept of a setvar variable not occurring in a wff (as opposed to just two setvar variables being distinct). We retain ax-c16 36643 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-5 1918, which might be easier to study for some theoretical purposes.

This axiom is obsolete and should no longer be used. It is proved above as Theorem axc16 2258. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-c16	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Detailed syntax breakdown of Axiom ax-c16

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2		vy	. . . 4 setvar 𝑦
3	1, 2	weq 1971	. . 3 wff 𝑥 = 𝑦
4	3, 1	wal 1541	. 2 wff ∀𝑥 𝑥 = 𝑦
5		wph	. . 3 wff 𝜑
6	5, 1	wal 1541	. . 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:  ax5eq  36683  axc16g-o  36685  ax5el  36688  axc11n-16  36689