Theorem bj-hbexd 37706

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

Hypotheses

Ref	Expression
bj-hbexd.nf	⊢ (𝜑 → ∀𝑦𝜓)
bj-hbexd.maj	⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))

Assertion

Ref	Expression
bj-hbexd	⊢ (𝜑 → (∃𝑦𝜒 → ∀𝑥∃𝑦𝜃))

Proof of Theorem bj-hbexd

Step	Hyp	Ref	Expression
1		bj-hbexd.nf	. 2 ⊢ (𝜑 → ∀𝑦𝜓)
2		19.12 2357	. . 3 ⊢ (∃𝑦∀𝑥𝜃 → ∀𝑥∃𝑦𝜃)
3	2	a1i 11	. 2 ⊢ (𝜑 → (∃𝑦∀𝑥𝜃 → ∀𝑥∃𝑦𝜃))
4		bj-hbexd.maj	. 2 ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃))
5	1, 3, 4	bj-exlimd 37601	1 ⊢ (𝜑 → (∃𝑦𝜒 → ∀𝑥∃𝑦𝜃))

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-hbext  37707