Axiom ax-ial 1580

Description: 𝑥 is not free in ∀𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

Assertion

Ref	Expression
ax-ial	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

Detailed syntax breakdown of Axiom ax-ial

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3	1, 2	wal 1393	. 2 wff ∀𝑥𝜑
4	3, 2	wal 1393	. 2 wff ∀𝑥∀𝑥𝜑
5	3, 4	wi 4	1 wff (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

This axiom is referenced by:  hba1  1586  ax6b  1697  exdistrfor  1846  sbcof2  1856  hbs1  1989  sbcomxyyz  2023