Theorem bj-cbvalimd 37530

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

Hypotheses

Ref	Expression
bj-cbvalimd.nf0	⊢ (𝜑 → ∀𝑥𝜑)
bj-cbvalimd.nf1	⊢ (𝜑 → ∀𝑦𝜑)
bj-cbvalimd.nfch	⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))
bj-cbvalimd.nfth	⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))
bj-cbvalimd.denote	⊢ (𝜑 → ∀𝑦∃𝑥𝜓)
bj-cbvalimd.maj	⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))

Assertion

Ref	Expression
bj-cbvalimd	⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))

Proof of Theorem bj-cbvalimd

Step	Hyp	Ref	Expression
1		bj-cbvalimd.nf0	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-cbvalimd.nf1	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-cbvalimd.nfch	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))
4	1, 3	hbald 2204	. 2 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦∀𝑥𝜒))
5		bj-cbvalimd.nfth	. 2 ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))
6		bj-cbvalimd.denote	. 2 ⊢ (𝜑 → ∀𝑦∃𝑥𝜓)
7		bj-cbvalimd.maj	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
8	1, 2, 4, 5, 6, 7	bj-cbvalimdlem 37528	1 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀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-11 2193

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by: (None)