Theorem mo4f 2001

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		ax-17 1459	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	mo3h 1994	. 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 1714	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	anbi2i 444	. . . 4 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ 𝜓))
7	6	imbi1i 236	. . 3 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
8	7	2albii 1400	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
9	2, 8	bitri 182	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282  Ⅎwnf 1389  [wsb 1685  ∃*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  mo4  2002  mob2  2772  moop2  4006  dffun4f  4938