Theorem wl-euequ1f 32535

Description: euequ1 2464 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)

Assertion

Ref	Expression
wl-euequ1f	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Proof of Theorem wl-euequ1f

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1877	. . 3 ⊢ ∃𝑧 𝑧 = 𝑦
2		nfv 1830	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
3		nfnae 2306	. . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4		nfeqf2 2285	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
5		equequ2 1940	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
6	5	equcoms 1934	. . . . . 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 2070	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧)))
9	2, 8	eximd 2072	. . 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		df-eu 2462	. 2 ⊢ (∃!𝑥 𝑥 = 𝑦 ↔ ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
12	10, 11	sylibr 223	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  ∃!weu 2458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462

This theorem is referenced by: (None)