Theorem mo4 2237

Description: "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Hypothesis

Ref	Expression
mo4.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
mo4	⊢ (∃*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 mo4

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxψ
2		mo4.1	. 2 ⊢ (x = y → (φ ↔ ψ))
3	1, 2	mo4f 2236	1 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃*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:  eu4  2243  rmo4  3030  dffun3  5121  dff13  5472  caovmo  5646  xpassen  6058  enpw1  6063