Theorem bj-hbex 37708

Description: A more general instance of hbex 2355. (Contributed by BJ, 4-Apr-2026.)

Hypothesis

Ref	Expression
bj-hbex.1	⊢ (𝜑 → ∀𝑥𝜓)

Assertion

Ref	Expression
bj-hbex	⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓)

Proof of Theorem bj-hbex

Step	Hyp	Ref	Expression
1		19.12 2357	. 2 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓)
2		bj-hbex.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3	1, 2	bj-sylge 37600	1 ⊢ (∃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: (None)