Theorem bj-hbext 37317

Description: Closed form of bj-hbex 37318 and hbex 2358. (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 2219	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓))
3	1, 2	bj-hbexd 37316	1 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1810  df-nf 1814

This theorem is referenced by:  bj-nfext  37320