Theorem eu4 2243

Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Hypothesis

Ref	Expression
eu4.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
eu4	⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ ψ) → x = y)))

Distinct variable groups:   x,y   φ,y   ψ,x

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem eu4

Step	Hyp	Ref	Expression
1		eu5 2242	. 2 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃*xφ))
2		eu4.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
3	2	mo4 2237	. . 3 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))
4	3	anbi2i 675	. 2 ⊢ ((∃xφ ∧ ∃*xφ) ↔ (∃xφ ∧ ∀x∀y((φ ∧ ψ) → x = y)))
5	1, 4	bitri 240	1 ⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ ψ) → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205

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:  euequ1  2292  eueq  3008  euind  3023  uniintsn  3963