Theorem reusv1 5015

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 3079	. . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
2	1	nfmo 2624	. . 3 ⊢ Ⅎ𝑦∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
3		rsp 3067	. . . . . 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 2006	. . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∀𝑥(∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶)))
6		mo2icl 3526	. . . 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 3162	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
9		mormo 3297	. 2 ⊢ (∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
10		reu5 3298	. . 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 985	. 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 196  ∀wal 1630   = wceq 1632   ∈ wcel 2139  ∃*wmo 2608  ∀wral 3050  ∃wrex 3051  ∃!wreu 3052  ∃*wrmo 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-v 3342

This theorem is referenced by:  cdleme25c  36145  cdleme29c  36166  cdlemefrs29cpre1  36188  cdlemk29-3  36701  cdlemkid5  36725  dihlsscpre  37025  mapdh9a  37581  mapdh9aOLDN  37582