Theorem bj-modal4e 34044

Description: First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 34043 (hba1 2297). (Contributed by BJ, 21-Dec-2020.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-modal4e

Step	Hyp	Ref	Expression
1		bj-modal4 34043	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑥 ¬ 𝜑)
2		alnex 1778	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3		2exnaln 1825	. . . 4 ⊢ (∃𝑥∃𝑥𝜑 ↔ ¬ ∀𝑥∀𝑥 ¬ 𝜑)
4	3	con2bii 360	. . 3 ⊢ (∀𝑥∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥∃𝑥𝜑)
5	1, 2, 4	3imtr3i 293	. 2 ⊢ (¬ ∃𝑥𝜑 → ¬ ∃𝑥∃𝑥𝜑)
6	5	con4i 114	1 ⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2172

This theorem depends on definitions:  df-bi 209  df-ex 1777

This theorem is referenced by:  bj-wnf1  34046  bj-nnfe1  34084