Axiom ax-bdal 13005

Description: A bounded universal quantification of a bounded formula is bounded. Note the disjoint variable condition on 𝑥, 𝑦. (Contributed by BJ, 25-Sep-2019.)

Hypothesis

Ref	Expression
bdal.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
ax-bdal	⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Detailed syntax breakdown of Axiom ax-bdal

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		vx	. . 3 setvar 𝑥
3		vy	. . . 4 setvar 𝑦
4	3	cv 1330	. . 3 class 𝑦
5	1, 2, 4	wral 2414	. 2 wff ∀𝑥 ∈ 𝑦 𝜑
6	5	wbd 12999	1 wff BOUNDED ∀𝑥 ∈ 𝑦 𝜑

This axiom is referenced by:  bdreu  13042  bdss  13051  bdcint  13064  bdciin  13066  bdcriota  13070  bj-bdind  13117  bj-nntrans  13138