Theorem hbald 2206

Description: Deduction form of bound-variable hypothesis builder hbal 2205. (Contributed by NM, 2-Jan-2002.)

Hypotheses

Ref	Expression
hbald.1	⊢ (𝜑 → ∀𝑦𝜑)
hbald.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
hbald	⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))

Proof of Theorem hbald

Step	Hyp	Ref	Expression
1		hbald.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2		hbald.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	1, 2	alimdh 1850	. 2 ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑦∀𝑥𝜓))
4		ax-11 2195	. 2 ⊢ (∀𝑦∀𝑥𝜓 → ∀𝑥∀𝑦𝜓)
5	3, 4	syl6 36	1 ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1828  ax-4 1842  ax-11 2195

This theorem is used by:  bj-cbvalimd0  37291  bj-cbvalimd  37294  bj-nnf-cbvali  37445  dvelimf-o  39744