Theorem rmo2 3887

Description: Alternate definition of restricted "at most one". Note that ∃*𝑥 ∈ 𝐴𝜑 is not equivalent to ∃𝑦 ∈ 𝐴∀𝑥 ∈ 𝐴(𝜑 → 𝑥 = 𝑦) (in analogy to reu6 3732); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3888. (Contributed by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
rmo2	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem rmo2

Step	Hyp	Ref	Expression
1		df-rmo 3380	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		nfv 1914	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3		rmo2.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4	2, 3	nfan 1899	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
5	4	mof 2563	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦))
6		impexp 450	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
7	6	albii 1819	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
8		df-ral 3062	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝑦)))
9	7, 8	bitr4i 278	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
10	9	exbii 1848	. 2 ⊢ (∃𝑦∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))
11	1, 5, 10	3bitri 297	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2108  ∃*wmo 2538  ∀wral 3061  ∃*wrmo 3379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-ral 3062  df-rmo 3380

This theorem is referenced by:  rmo2i  3888  rmoanimALT  3895  disjiun  5131  poimirlem2  37629  onsucf1lem  43282