Theorem bj-exexalal 36831

Description: A lemma for changing bound variables. Only the forward implication is intuitionistic. (Contributed by BJ, 14-Mar-2026.)

Assertion

Ref	Expression
bj-exexalal	⊢ ((∃𝑥𝜑 → ∃𝑦𝜓) ↔ (∀𝑦 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Proof of Theorem bj-exexalal

Step	Hyp	Ref	Expression
1		con34b 316	. 2 ⊢ ((∃𝑥𝜑 → ∃𝑦𝜓) ↔ (¬ ∃𝑦𝜓 → ¬ ∃𝑥𝜑))
2		alnex 1783	. . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
3		alnex 1783	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
4	2, 3	imbi12i 350	. 2 ⊢ ((∀𝑦 ¬ 𝜓 → ∀𝑥 ¬ 𝜑) ↔ (¬ ∃𝑦𝜓 → ¬ ∃𝑥𝜑))
5	1, 4	bitr4i 278	1 ⊢ ((∃𝑥𝜑 → ∃𝑦𝜓) ↔ (∀𝑦 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by:  bj-cbvaew  36880