Theorem euequ1 2292

Description: Equality has existential uniqueness. Special case of eueq1 3010 proved using only predicate calculus. (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		a9ev 1656	. 2 ⊢ ∃x x = y
2		equtr2 1688	. . 3 ⊢ ((x = y ∧ z = y) → x = z)
3	2	gen2 1547	. 2 ⊢ ∀x∀z((x = y ∧ z = y) → x = z)
4		equequ1 1684	. . 3 ⊢ (x = z → (x = y ↔ z = y))
5	4	eu4 2243	. 2 ⊢ (∃!x x = y ↔ (∃x x = y ∧ ∀x∀z((x = y ∧ z = y) → x = z)))
6	1, 3, 5	mpbir2an 886	1 ⊢ ∃!x x = y

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  copsexg  4608  oprabid  5551  scancan  6332