Theorem euequ1 2182

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 1748	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		equtr2 1763	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)
3	2	gen2 1503	. 2 ⊢ ∀𝑥∀𝑧((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)
4		equequ1 1764	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
5	4	eu4 2149	. 2 ⊢ (∃!𝑥 𝑥 = 𝑦 ↔ (∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∀𝑧((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑧)))
6	1, 3, 5	mpbir2an 955	1 ⊢ ∃!𝑥 𝑥 = 𝑦

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400  ∃wex 1545  ∃!weu 2086

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by:  copsexg  4382  oprabid  6111