Theorem reusv1 5355

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 3262	. . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
2	1	nfmov 2554	. . 3 ⊢ Ⅎ𝑦∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
3		rsp 3226	. . . . . 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 1929	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∀𝑥(∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶)))
6		mo2icl 3688	. . . 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 3238	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
9		mormo 3361	. 2 ⊢ (∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
10		reu5 3358	. . 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 538	. 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 206  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2532  ∀wral 3045  ∃wrex 3054  ∃!wreu 3354  ∃*wrmo 3355

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-v 3452

This theorem is referenced by:  cdleme25c  40356  cdleme29c  40377  cdlemefrs29cpre1  40399  cdlemk29-3  40912  cdlemkid5  40936  dihlsscpre  41235  mapdh9a  41790  mapdh9aOLDN  41791