Theorem euanv 2265

Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)

Assertion

Ref	Expression
euanv	⊢ (∃!x(φ ∧ ψ) ↔ (φ ∧ ∃!xψ))

Distinct variable group:   φ,x

Allowed substitution hint:   ψ(x)

Proof of Theorem euanv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2	1	euan 2261	1 ⊢ (∃!x(φ ∧ ψ) ↔ (φ ∧ ∃!xψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃!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:  eueq2  3010  2reu5lem1  3041  fsn  5432