Theorem rmo4f 3728

Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)

Hypotheses

Ref	Expression
rmo4f.1	⊢ Ⅎ𝑥𝐴
rmo4f.2	⊢ Ⅎ𝑦𝐴
rmo4f.3	⊢ Ⅎ𝑥𝜓
rmo4f.4	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rmo4f	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

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

Proof of Theorem rmo4f

Step	Hyp	Ref	Expression
1		rmo4f.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		rmo4f.2	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1915	. . 3 ⊢ Ⅎ𝑦𝜑
4	1, 2, 3	rmo3f 3727	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
5		rmo4f.3	. . . . . 6 ⊢ Ⅎ𝑥𝜓
6		rmo4f.4	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
7	5, 6	sbiev 2330	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
8	7	anbi2i 624	. . . 4 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ 𝜓))
9	8	imbi1i 352	. . 3 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
10	9	2ralbii 3168	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
11	4, 10	bitri 277	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  Ⅎwnf 1784  [wsb 2069  Ⅎwnfc 2963  ∀wral 3140  ∃*wrmo 3143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-clel 2895  df-nfc 2965  df-ral 3145  df-rmo 3148

This theorem is referenced by:  2sqreulem4  26032  disjorf  30331  funcnv5mpt  30415