Theorem bj-hbext 37198

Description: Closed form of bj-hbex 37199 and hbex 2360. (Contributed by BJ, 10-Oct-2019.)

Assertion

Ref	Expression
bj-hbext	⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))

Proof of Theorem bj-hbext

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓))
2		sp 2221	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓))
3	1, 2	bj-hbexd 37197	1 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-11 2194  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1803  df-nf 1807

This theorem is referenced by:  bj-nfext  37201