Theorem bj-exexalal 37227

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 319	. 2 ⊢ ((∃𝑥𝜑 → ∃𝑦𝜓) ↔ (¬ ∃𝑦𝜓 → ¬ ∃𝑥𝜑))
2		alnex 1810	. . 3 ⊢ (∀𝑦 ¬ 𝜓 ↔ ¬ ∃𝑦𝜓)
3		alnex 1810	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
4	2, 3	imbi12i 353	. 2 ⊢ ((∀𝑦 ¬ 𝜓 → ∀𝑥 ¬ 𝜑) ↔ (¬ ∃𝑦𝜓 → ¬ ∃𝑥𝜑))
5	1, 4	bitr4i 281	1 ⊢ ((∃𝑥𝜑 → ∃𝑦𝜓) ↔ (∀𝑦 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ex 1809

This theorem is used by:  bj-cbvaew  37294