Theorem modc 1943

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 1941	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3		exmiddc 744	. . 3 ⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
4	1	mor 1942	. . . 4 ⊢ (∃𝑥𝜑 → (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
5	1	mo2n 1928	. . . . 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 636	. . 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 131	1 ⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646

This theorem is referenced by:  mo2dc  1955