Theorem bj-hbald 36944

Description: General statement that hbald 2174 proves . (Contributed by BJ, 4-Apr-2026.)

Hypotheses

Ref	Expression
bj-hbald.1	⊢ (𝜑 → ∀𝑦𝜓)
bj-hbald.2	⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))

Assertion

Ref	Expression
bj-hbald	⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥∀𝑦𝜃))

Proof of Theorem bj-hbald

Step	Hyp	Ref	Expression
1		bj-hbald.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜓)
2		bj-hbald.2	. . . 4 ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))
3	2	al2imi 1817	. . 3 ⊢ (∀𝑦𝜓 → (∀𝑦𝜒 → ∀𝑦∀𝑥𝜃))
4	1, 3	syl 17	. 2 ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑦∀𝑥𝜃))
5		ax-11 2163	. 2 ⊢ (∀𝑦∀𝑥𝜃 → ∀𝑥∀𝑦𝜃)
6	4, 5	syl6 35	1 ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥∀𝑦𝜃))

Syntax hints:   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1797  ax-4 1811  ax-11 2163

This theorem is referenced by:  bj-hbalt  36945