Theorem eunexOLD 5138

Description: Obsolete proof of eunex 5137 as of 2-Jan-2023. (Contributed by NM, 24-Oct-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
eunexOLD	⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Proof of Theorem eunexOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dtru 5118	. . . . 5 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2		alim 1773	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∀𝑥𝜑 → ∀𝑥 𝑥 = 𝑦))
3	1, 2	mtoi 191	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
4	3	exlimiv 1889	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
5	4	adantl 474	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ¬ ∀𝑥𝜑)
6		eu3v 2581	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
7		exnal 1789	. 2 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
8	5, 6, 7	3imtr4i 284	1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387  ∀wal 1505  ∃wex 1742  ∃!weu 2579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-12 2104  ax-nul 5061  ax-pow 5113

This theorem depends on definitions:  df-bi 199  df-an 388  df-tru 1510  df-ex 1743  df-nf 1747  df-mo 2544  df-eu 2580

This theorem is referenced by: (None)