Theorem exexw 2050

Description: Existential quantification over a given variable is idempotent. Weak version of bj-exexbiex 36702, requiring fewer axioms. (Contributed by GG, 4-Nov-2024.)

Hypothesis

Ref	Expression
exexw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
exexw	⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑥𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem exexw

Step	Hyp	Ref	Expression
1		exexw.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	notbid 318	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	hba1w 2046	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑥 ¬ 𝜑)
4	2	spw 2032	. . . . 5 ⊢ (∀𝑥 ¬ 𝜑 → ¬ 𝜑)
5	4	alimi 1810	. . . 4 ⊢ (∀𝑥∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ 𝜑)
6	3, 5	impbii 209	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ∀𝑥∀𝑥 ¬ 𝜑)
7	6	notbii 320	. 2 ⊢ (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥∀𝑥 ¬ 𝜑)
8		df-ex 1779	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
9		2exnaln 1828	. 2 ⊢ (∃𝑥∃𝑥𝜑 ↔ ¬ ∀𝑥∀𝑥 ¬ 𝜑)
10	7, 8, 9	3bitr4i 303	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by:  dfid2  5579  bj-dfid2ALT  37067