Theorem bj-cbvaew 37080

Description: Exixtentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37076. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cbvaew	⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∃𝑦𝜓 → ∃𝑥𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-cbvaew

Step	Hyp	Ref	Expression
1		notnotb 317	. . . . 5 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1838	. . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		df-fal 1572	. . . . 5 ⊢ (⊥ ↔ ¬ ⊤)
4	3	albii 1838	. . . 4 ⊢ (∀𝑦⊥ ↔ ∀𝑦 ¬ ⊤)
5	2, 4	imbi12i 352	. . 3 ⊢ ((∀𝑥𝜑 → ∀𝑦⊥) ↔ (∀𝑥 ¬ ¬ 𝜑 → ∀𝑦 ¬ ⊤))
6		bj-exexalal 37013	. . 3 ⊢ ((∃𝑦⊤ → ∃𝑥 ¬ 𝜑) ↔ (∀𝑥 ¬ ¬ 𝜑 → ∀𝑦 ¬ ⊤))
7	5, 6	bitr4i 280	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑦⊥) ↔ (∃𝑦⊤ → ∃𝑥 ¬ 𝜑))
8		bj-cbvew 37078	. 2 ⊢ ((∃𝑦⊤ → ∃𝑥 ¬ 𝜑) → (∃𝑦𝜓 → ∃𝑥𝜓))
9	7, 8	sylbi 219	1 ⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∃𝑦𝜓 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1557  ⊤wtru 1560  ⊥wfal 1571  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799

This theorem is referenced by: (None)