Theorem bj-cbvalimd0 37105

Description: A lemma for alpha-renaming of variables bound by a universal quantifier. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1991 will prove Hypothesis bj-cbvalimd0.denote. When ax6ev 1991 is not available but only its universal closure is, then bj-cbvalimd 37108 or bj-cbvalimdv 37110 should be used (see bj-cbvalimdlem 37106, bj-cbval 37123). (Contributed by BJ, 4-Apr-2026.)

Hypotheses

Ref	Expression
bj-cbvalimd0.nf0	⊢ (𝜑 → ∀𝑥𝜑)
bj-cbvalimd0.nf1	⊢ (𝜑 → ∀𝑦𝜑)
bj-cbvalimd0.nfch	⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))
bj-cbvalimd0.nfth	⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))
bj-cbvalimd0.denote	⊢ (𝜑 → ∃𝑥𝜓)
bj-cbvalimd0.maj	⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))

Assertion

Ref	Expression
bj-cbvalimd0	⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))

Proof of Theorem bj-cbvalimd0

Step	Hyp	Ref	Expression
1		bj-cbvalimd0.nf1	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
2		bj-cbvalimd0.nf0	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		bj-cbvalimd0.nfch	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))
4	2, 3	hbald 2204	. 2 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦∀𝑥𝜒))
5		bj-cbvalimd0.nfth	. . 3 ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))
6		bj-cbvalimd0.denote	. . 3 ⊢ (𝜑 → ∃𝑥𝜓)
7		bj-cbvalimd0.maj	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
8	2, 5, 6, 7	bj-spim 37103	. 2 ⊢ (𝜑 → (∀𝑥𝜒 → 𝜃))
9	1, 4, 8	bj-alrimd 37073	1 ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1560  ∃wex 1801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-11 2193

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802

This theorem is referenced by: (None)