Theorem bj-cbveximd 36894

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

Assertion

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

Proof of Theorem bj-cbveximd

Step	Hyp	Ref	Expression
1		bj-cbveximd.nf0	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-cbveximd.nf1	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-cbveximd.nfch	. 2 ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒))
4		excomim 2169	. . 3 ⊢ (∃𝑥∃𝑦𝜃 → ∃𝑦∃𝑥𝜃)
5		bj-cbveximd.nfth	. . . 4 ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃))
6	2, 5	eximdh 1866	. . 3 ⊢ (𝜑 → (∃𝑦∃𝑥𝜃 → ∃𝑦𝜃))
7	4, 6	syl5 34	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦𝜃 → ∃𝑦𝜃))
8		bj-cbveximd.denote	. 2 ⊢ (𝜑 → ∀𝑥∃𝑦𝜓)
9		bj-cbveximd.maj	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
10	1, 2, 3, 7, 8, 9	bj-cbveximdlem 36892	1 ⊢ (𝜑 → (∃𝑥𝜒 → ∃𝑦𝜃))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-11 2163

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by: (None)