Theorem bj-hbex 37318

Description: A more general instance of hbex 2358. (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 2360	. 2 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓)
2		bj-hbex.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3	1, 2	bj-sylge 37210	1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-or 861  df-ex 1810  df-nf 1814

This theorem is referenced by: (None)