Theorem mo4f 2114

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 1549	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	mo3h 2107	. 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 1814	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	anbi2i 457	. . . 4 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ 𝜓))
7	6	imbi1i 238	. . 3 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
8	7	2albii 1494	. 2 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
9	2, 8	bitri 184	1 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371  Ⅎwnf 1483  [wsb 1785  ∃*wmo 2055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by:  mo4  2115  mob2  2953  moop2  4296  dffun4f  5287