Theorem modc 1984

Description: Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)

Hypothesis

Ref	Expression
modc.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
modc	⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem modc

Step	Hyp	Ref	Expression
1		modc.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	mo23 1982	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3		exmiddc 777	. . 3 ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1	mor 1983	. . . 4 ⊢ (∃𝑥𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
5	1	mo2n 1969	. . . . 5 ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
6	5	a1d 22	. . . 4 ⊢ (¬ ∃𝑥𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
7	4, 6	jaoi 668	. . 3 ⊢ ((∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑) → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
8	3, 7	syl 14	. 2 ⊢ (DECID ∃𝑥𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
9	2, 8	impbid2 141	1 ⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661  DECID wdc 775  ∀wal 1282  Ⅎwnf 1389  ∃wex 1421  [wsb 1685

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-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686

This theorem is referenced by:  mo2dc  1996