Theorem mo2dc 2142

Description: Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)

Hypothesis

Ref	Expression
mo2dc.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
mo2dc	⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo2dc

Step	Hyp	Ref	Expression
1		mo2dc.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1572	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	mo3h 2140	. 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4	1	modc 2130	. 2 ⊢ (DECID ∃𝑥𝜑 → (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
5	3, 4	bitr4id 199	1 ⊢ (DECID ∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846  ∀wal 1400  Ⅎwnf 1513  ∃wex 1545  [wsb 1815  ∃*wmo 2087

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by: (None)