Theorem euequ1 2114

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 1689	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		equtr2 1704	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)
3	2	gen2 1443	. 2 ⊢ ∀𝑥∀𝑧((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)
4		equequ1 1705	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
5	4	eu4 2081	. 2 ⊢ (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∀𝑧((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)))
6	1, 3, 5	mpbir2an 937	1 ⊢ ∃!𝑥 𝑥 = 𝑦

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1346  ∃wex 1485  ∃!weu 2019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by:  copsexg  4229  oprabid  5885