Theorem wl-euequf 36949

Description: euequ 2585 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)

Assertion

Ref	Expression
wl-euequf	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Proof of Theorem wl-euequf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1965	. . 3 ⊢ ∃𝑧 𝑧 = 𝑦
2		nfv 1909	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
3		nfna1 2141	. . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4		nfeqf2 2370	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
5		equequ2 2021	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
6	5	equcoms 2015	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
7	6	a1i 11	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧)))
8	3, 4, 7	alrimdd 2199	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧)))
9	2, 8	eximd 2201	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧)))
10	1, 9	mpi 20	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
11		eu6 2562	. 2 ⊢ (∃!𝑥 𝑥 = 𝑦 ↔ ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
12	10, 11	sylibr 233	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1531  ∃wex 1773  ∃!weu 2556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2365

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-mo 2528  df-eu 2557

This theorem is referenced by: (None)