Theorem bj-modal4e 37749

Description: First-order logic form of the modal axiom (4) using existential quantifiers. Dual statement of bj-modal4 37748 (hba1 2329). (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 37748	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑥 ¬ 𝜑)
2		alnex 1813	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3		2exnaln 1861	. . . 4 ⊢ (∃𝑥∃𝑥𝜑 ↔ ¬ ∀𝑥∀𝑥 ¬ 𝜑)
4	3	con2bii 360	. . 3 ⊢ (∀𝑥∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥∃𝑥𝜑)
5	1, 2, 4	3imtr3i 294	. 2 ⊢ (¬ ∃𝑥𝜑 → ¬ ∃𝑥∃𝑥𝜑)
6	5	con4i 115	1 ⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-10 2179  ax-12 2215

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  bj-wnf1  37751  bj-nnfe1  37817