Theorem hbald 2028

Description: Deduction form of bound-variable hypothesis builder hbal 2023. (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 1735	. 2 ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑦∀𝑥𝜓))
4		ax-11 2021	. 2 ⊢ (∀𝑦∀𝑥𝜓 → ∀𝑥∀𝑦𝜓)
5	3, 4	syl6 34	1 ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1713  ax-4 1728  ax-11 2021

This theorem is referenced by:  nfaldOLD  2152  dvelimf-o  33232