Theorem bj-hbex 37378

Description: A more general instance of hbex 2361. (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 2363	. 2 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓)
2		bj-hbex.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3	1, 2	bj-sylge 37270	1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by: (None)