Theorem reusv1 4156

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.)

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 2349	. . . 4 ⊢ Ⅎy∀y ∈ B (φ → x = 𝐶)
2	1	nfmo 1917	. . 3 ⊢ Ⅎy∃*x∀y ∈ B (φ → x = 𝐶)
3		rsp 2363	. . . . . . . 8 ⊢ (∀y ∈ B (φ → x = 𝐶) → (y ∈ B → (φ → x = 𝐶)))
4	3	impd 242	. . . . . . 7 ⊢ (∀y ∈ B (φ → x = 𝐶) → ((y ∈ B ∧ φ) → x = 𝐶))
5	4	com12 27	. . . . . 6 ⊢ ((y ∈ B ∧ φ) → (∀y ∈ B (φ → x = 𝐶) → x = 𝐶))
6	5	alrimiv 1751	. . . . 5 ⊢ ((y ∈ B ∧ φ) → ∀x(∀y ∈ B (φ → x = 𝐶) → x = 𝐶))
7		moeq 2710	. . . . 5 ⊢ ∃*x x = 𝐶
8		moim 1961	. . . . 5 ⊢ (∀x(∀y ∈ B (φ → x = 𝐶) → x = 𝐶) → (∃*x x = 𝐶 → ∃*x∀y ∈ B (φ → x = 𝐶)))
9	6, 7, 8	mpisyl 1332	. . . 4 ⊢ ((y ∈ B ∧ φ) → ∃*x∀y ∈ B (φ → x = 𝐶))
10	9	ex 108	. . 3 ⊢ (y ∈ B → (φ → ∃*x∀y ∈ B (φ → x = 𝐶)))
11	2, 10	rexlimi 2420	. 2 ⊢ (∃y ∈ B φ → ∃*x∀y ∈ B (φ → x = 𝐶))
12		mormo 2515	. 2 ⊢ (∃*x∀y ∈ B (φ → x = 𝐶) → ∃*x ∈ A ∀y ∈ B (φ → x = 𝐶))
13		reu5 2516	. . 3 ⊢ (∃!x ∈ A ∀y ∈ B (φ → x = 𝐶) ↔ (∃x ∈ A ∀y ∈ B (φ → x = 𝐶) ∧ ∃*x ∈ A ∀y ∈ B (φ → x = 𝐶)))
14	13	rbaib 829	. 2 ⊢ (∃*x ∈ A ∀y ∈ B (φ → x = 𝐶) → (∃!x ∈ A ∀y ∈ B (φ → x = 𝐶) ↔ ∃x ∈ A ∀y ∈ B (φ → x = 𝐶)))
15	11, 12, 14	3syl 17	1 ⊢ (∃y ∈ B φ → (∃!x ∈ A ∀y ∈ B (φ → x = 𝐶) ↔ ∃x ∈ A ∀y ∈ B (φ → x = 𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  ∃*wmo 1898  ∀wral 2300  ∃wrex 2301  ∃!wreu 2302  ∃*wrmo 2303

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-v 2553

This theorem is referenced by: (None)