Theorem bj-hbext 36976

Description: Closed form of bj-hbex 36977 and hbex 2331. (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 22	. 2 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → ∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓))
2		sp 2191	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓))
3	1, 2	bj-hbexd 36975	1 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1782  df-nf 1786

This theorem is referenced by:  bj-nfext  36979