Theorem bj-hbexd 37582

Description: A more general instance of the deduction form of hbex 2356. (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 2358	. . 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 37477	1 ⊢ (𝜑 → (∃𝑦𝜒 → ∀𝑥∃𝑦𝜃))

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 2178  ax-11 2194  ax-12 2213

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

This theorem is used by:  bj-hbext  37583