Theorem euequ1 2140

Description: Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)

Assertion

Ref	Expression
euequ1	⊢ ∃!𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem euequ1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1710	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		equtr2 1725	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)
3	2	gen2 1464	. 2 ⊢ ∀𝑥∀𝑧((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)
4		equequ1 1726	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
5	4	eu4 2107	. 2 ⊢ (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∀𝑧((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)))
6	1, 3, 5	mpbir2an 944	1 ⊢ ∃!𝑥 𝑥 = 𝑦

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1506  ∃!weu 2045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049

This theorem is referenced by:  copsexg  4277  oprabid  5954