Theorem bj-hbex 37151

Description: A more general instance of hbex 2356. (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 2358	. 2 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓)
2		bj-hbex.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3	1, 2	bj-sylge 37043	1 ⊢ (∃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: (None)