Theorem rexxfrd 4161

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)

Hypotheses

Ref	Expression
ralxfrd.1	⊢ ((φ ∧ y ∈ 𝐶) → A ∈ B)
ralxfrd.2	⊢ ((φ ∧ x ∈ B) → ∃y ∈ 𝐶 x = A)
ralxfrd.3	⊢ ((φ ∧ x = A) → (ψ ↔ χ))

Assertion

Ref	Expression
rexxfrd	⊢ (φ → (∃x ∈ B ψ ↔ ∃y ∈ 𝐶 χ))

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

Allowed substitution hints:   ψ(x)   χ(y)   A(y)   𝐶(y)

Proof of Theorem rexxfrd

Step	Hyp	Ref	Expression
1		nfv 1418	. . . . 5 ⊢ Ⅎyψ
2	1	19.3 1443	. . . 4 ⊢ (∀yψ ↔ ψ)
3		ralxfrd.2	. . . . 5 ⊢ ((φ ∧ x ∈ B) → ∃y ∈ 𝐶 x = A)
4		df-rex 2306	. . . . . . . 8 ⊢ (∃y ∈ 𝐶 x = A ↔ ∃y(y ∈ 𝐶 ∧ x = A))
5		19.29 1508	. . . . . . . . . 10 ⊢ ((∀yψ ∧ ∃y(y ∈ 𝐶 ∧ x = A)) → ∃y(ψ ∧ (y ∈ 𝐶 ∧ x = A)))
6		an12 495	. . . . . . . . . . 11 ⊢ ((ψ ∧ (y ∈ 𝐶 ∧ x = A)) ↔ (y ∈ 𝐶 ∧ (ψ ∧ x = A)))
7	6	exbii 1493	. . . . . . . . . 10 ⊢ (∃y(ψ ∧ (y ∈ 𝐶 ∧ x = A)) ↔ ∃y(y ∈ 𝐶 ∧ (ψ ∧ x = A)))
8	5, 7	sylib 127	. . . . . . . . 9 ⊢ ((∀yψ ∧ ∃y(y ∈ 𝐶 ∧ x = A)) → ∃y(y ∈ 𝐶 ∧ (ψ ∧ x = A)))
9		df-rex 2306	. . . . . . . . 9 ⊢ (∃y ∈ 𝐶 (ψ ∧ x = A) ↔ ∃y(y ∈ 𝐶 ∧ (ψ ∧ x = A)))
10	8, 9	sylibr 137	. . . . . . . 8 ⊢ ((∀yψ ∧ ∃y(y ∈ 𝐶 ∧ x = A)) → ∃y ∈ 𝐶 (ψ ∧ x = A))
11	4, 10	sylan2b 271	. . . . . . 7 ⊢ ((∀yψ ∧ ∃y ∈ 𝐶 x = A) → ∃y ∈ 𝐶 (ψ ∧ x = A))
12		ralxfrd.3	. . . . . . . . . . 11 ⊢ ((φ ∧ x = A) → (ψ ↔ χ))
13	12	biimpd 132	. . . . . . . . . 10 ⊢ ((φ ∧ x = A) → (ψ → χ))
14	13	expimpd 345	. . . . . . . . 9 ⊢ (φ → ((x = A ∧ ψ) → χ))
15	14	ancomsd 256	. . . . . . . 8 ⊢ (φ → ((ψ ∧ x = A) → χ))
16	15	reximdv 2414	. . . . . . 7 ⊢ (φ → (∃y ∈ 𝐶 (ψ ∧ x = A) → ∃y ∈ 𝐶 χ))
17	11, 16	syl5 28	. . . . . 6 ⊢ (φ → ((∀yψ ∧ ∃y ∈ 𝐶 x = A) → ∃y ∈ 𝐶 χ))
18	17	adantr 261	. . . . 5 ⊢ ((φ ∧ x ∈ B) → ((∀yψ ∧ ∃y ∈ 𝐶 x = A) → ∃y ∈ 𝐶 χ))
19	3, 18	mpan2d 404	. . . 4 ⊢ ((φ ∧ x ∈ B) → (∀yψ → ∃y ∈ 𝐶 χ))
20	2, 19	syl5bir 142	. . 3 ⊢ ((φ ∧ x ∈ B) → (ψ → ∃y ∈ 𝐶 χ))
21	20	rexlimdva 2427	. 2 ⊢ (φ → (∃x ∈ B ψ → ∃y ∈ 𝐶 χ))
22		ralxfrd.1	. . . 4 ⊢ ((φ ∧ y ∈ 𝐶) → A ∈ B)
23	12	adantlr 446	. . . 4 ⊢ (((φ ∧ y ∈ 𝐶) ∧ x = A) → (ψ ↔ χ))
24	22, 23	rspcedv 2654	. . 3 ⊢ ((φ ∧ y ∈ 𝐶) → (χ → ∃x ∈ B ψ))
25	24	rexlimdva 2427	. 2 ⊢ (φ → (∃y ∈ 𝐶 χ → ∃x ∈ B ψ))
26	21, 25	impbid 120	1 ⊢ (φ → (∃x ∈ B ψ ↔ ∃y ∈ 𝐶 χ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301

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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553

This theorem is referenced by:  rexxfr2d  4163  rexxfr  4166