Theorem bj-hbext 37150

Description: Closed form of bj-hbex 37151 and hbex 2356. (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 2217	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓))
3	1, 2	bj-hbexd 37149	1 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-or 859  df-ex 1799  df-nf 1803

This theorem is referenced by:  bj-nfext  37153