Theorem bj-cbvalimdv 36988

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

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜒,𝑦

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

Proof of Theorem bj-cbvalimdv

Step	Hyp	Ref	Expression
1		bj-cbvalimdv.nf0	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-cbvalimdv.nf1	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		ax5d 1919	. 2 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦∀𝑥𝜒))
4		bj-cbvalimdv.nfth	. 2 ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))
5		bj-cbvalimdv.denote	. 2 ⊢ (𝜑 → ∀𝑦∃𝑥𝜓)
6		bj-cbvalimdv.maj	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
7	1, 2, 3, 4, 5, 6	bj-cbvalimdlem 36984	1 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1546  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788

This theorem is referenced by:  bj-cbval  37001