Theorem bj-cbveximdv 37297

Description: A lemma for alpha-renaming of variables bound by an existential quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbveximdv.nf0	⊢ (𝜑 → ∀𝑥𝜑)
bj-cbveximdv.nf1	⊢ (𝜑 → ∀𝑦𝜑)
bj-cbveximdv.nfth	⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))
bj-cbveximdv.denote	⊢ (𝜑 → ∀𝑥∃𝑦𝜓)
bj-cbveximdv.maj	⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))

Assertion

Ref	Expression
bj-cbveximdv	⊢ (𝜑 → (∃𝑥𝜒 → ∃𝑦𝜃))

Distinct variable groups:   𝑥,𝑦   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)   𝜃(𝑦)

Proof of Theorem bj-cbveximdv

Step	Hyp	Ref	Expression
1		bj-cbveximdv.nf0	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-cbveximdv.nf1	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-cbveximdv.nfth	. 2 ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))
4		ax5e 1945	. . 3 ⊢ (∃𝑥∃𝑦𝜃 → ∃𝑦𝜃)
5	4	a1i 11	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦𝜃 → ∃𝑦𝜃))
6		bj-cbveximdv.denote	. 2 ⊢ (𝜑 → ∀𝑥∃𝑦𝜓)
7		bj-cbveximdv.maj	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
8	1, 2, 3, 5, 6, 7	bj-cbveximdlem 37293	1 ⊢ (𝜑 → (∃𝑥𝜒 → ∃𝑦𝜃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀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

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  bj-cbvex  37310