Theorem bj-hbext 37707

Description: Closed form of bj-hbex 37708 and hbex 2355. (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 37706	1 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜓) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by:  bj-nfext  37710