Theorem bj-modal4e 32705

Description: Dual statement of hba1 2151 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.)

Assertion

Ref	Expression
bj-modal4e	⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)

Proof of Theorem bj-modal4e

Step	Hyp	Ref	Expression
1		hba1 2151	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑥 ¬ 𝜑)
2		alnex 1706	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3		2exnaln 1756	. . . 4 ⊢ (∃𝑥∃𝑥𝜑 ↔ ¬ ∀𝑥∀𝑥 ¬ 𝜑)
4	3	con2bii 347	. . 3 ⊢ (∀𝑥∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥∃𝑥𝜑)
5	1, 2, 4	3imtr3i 280	. 2 ⊢ (¬ ∃𝑥𝜑 → ¬ ∃𝑥∃𝑥𝜑)
6	5	con4i 113	1 ⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)