Theorem mo4f 2236

Description: "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Hypotheses

Ref	Expression
mo4f.1	⊢ Ⅎxψ
mo4f.2	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
mo4f	⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))

Distinct variable groups:   x,y   φ,y

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

Proof of Theorem mo4f

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎyφ
2	1	mo3 2235	. 2 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
3		mo4f.1	. . . . . 6 ⊢ Ⅎxψ
4		mo4f.2	. . . . . 6 ⊢ (x = y → (φ ↔ ψ))
5	3, 4	sbie 2038	. . . . 5 ⊢ ([y / x]φ ↔ ψ)
6	5	anbi2i 675	. . . 4 ⊢ ((φ ∧ [y / x]φ) ↔ (φ ∧ ψ))
7	6	imbi1i 315	. . 3 ⊢ (((φ ∧ [y / x]φ) → x = y) ↔ ((φ ∧ ψ) → x = y))
8	7	2albii 1567	. 2 ⊢ (∀x∀y((φ ∧ [y / x]φ) → x = y) ↔ ∀x∀y((φ ∧ ψ) → x = y))
9	2, 8	bitri 240	1 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ ψ) → x = y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544  [wsb 1648  ∃*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:  mo4  2237  mob2  3017