Theorem reusv1 5320

Description: Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)

Assertion

Ref	Expression
reusv1	⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem reusv1

Step	Hyp	Ref	Expression
1		nfra1 3144	. . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
2	1	nfmov 2560	. . 3 ⊢ Ⅎ𝑦∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
3		rsp 3131	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
4	3	com3l 89	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶)))
5	4	alrimdv 1932	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∀𝑥(∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶)))
6		mo2icl 3649	. . . 4 ⊢ (∀𝑥(∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶) → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
7	5, 6	syl6 35	. . 3 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
8	2, 7	rexlimi 3248	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
9		mormo 3360	. 2 ⊢ (∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
10		reu5 3361	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
11	10	rbaib 539	. 2 ⊢ (∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
12	8, 9, 11	3syl 18	1 ⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∃*wmo 2538  ∀wral 3064  ∃wrex 3065  ∃!wreu 3066  ∃*wrmo 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-v 3434

This theorem is referenced by:  cdleme25c  38369  cdleme29c  38390  cdlemefrs29cpre1  38412  cdlemk29-3  38925  cdlemkid5  38949  dihlsscpre  39248  mapdh9a  39803  mapdh9aOLDN  39804