Theorem bj-cbvaew 36880

Description: Exixtentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 36876. (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 315	. . . . 5 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1821	. . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		df-fal 1555	. . . . 5 ⊢ (⊥ ↔ ¬ ⊤)
4	3	albii 1821	. . . 4 ⊢ (∀𝑦⊥ ↔ ∀𝑦 ¬ ⊤)
5	2, 4	imbi12i 350	. . 3 ⊢ ((∀𝑥𝜑 → ∀𝑦⊥) ↔ (∀𝑥 ¬ ¬ 𝜑 → ∀𝑦 ¬ ⊤))
6		bj-exexalal 36831	. . 3 ⊢ ((∃𝑦⊤ → ∃𝑥 ¬ 𝜑) ↔ (∀𝑥 ¬ ¬ 𝜑 → ∀𝑦 ¬ ⊤))
7	5, 6	bitr4i 278	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑦⊥) ↔ (∃𝑦⊤ → ∃𝑥 ¬ 𝜑))
8		bj-cbvew 36878	. 2 ⊢ ((∃𝑦⊤ → ∃𝑥 ¬ 𝜑) → (∃𝑦𝜓 → ∃𝑥𝜓))
9	7, 8	sylbi 217	1 ⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∃𝑦𝜓 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ⊤wtru 1543  ⊥wfal 1554  ∃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-5 1912

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782

This theorem is referenced by: (None)