Theorem rexxfrd 4145

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 1402	. . . . 5 ⊢ Ⅎyψ
2	1	19.3 1428	. . . 4 ⊢ (∀yψ ↔ ψ)
3		ralxfrd.2	. . . . 5 ⊢ ((φ ∧ x ∈ B) → ∃y ∈ 𝐶 x = A)
4		df-rex 2290	. . . . . . . 8 ⊢ (∃y ∈ 𝐶 x = A ↔ ∃y(y ∈ 𝐶 ∧ x = A))
5		19.29 1493	. . . . . . . . . 10 ⊢ ((∀yψ ∧ ∃y(y ∈ 𝐶 ∧ x = A)) → ∃y(ψ ∧ (y ∈ 𝐶 ∧ x = A)))
6		an12 483	. . . . . . . . . . 11 ⊢ ((ψ ∧ (y ∈ 𝐶 ∧ x = A)) ↔ (y ∈ 𝐶 ∧ (ψ ∧ x = A)))
7	6	exbii 1478	. . . . . . . . . 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 2290	. . . . . . . . 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 2398	. . . . . . 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 406	. . . 4 ⊢ ((φ ∧ x ∈ B) → (∀yψ → ∃y ∈ 𝐶 χ))
20	2, 19	syl5bir 142	. . 3 ⊢ ((φ ∧ x ∈ B) → (ψ → ∃y ∈ 𝐶 χ))
21	20	rexlimdva 2411	. 2 ⊢ (φ → (∃x ∈ B ψ → ∃y ∈ 𝐶 χ))
22		ralxfrd.1	. . . 4 ⊢ ((φ ∧ y ∈ 𝐶) → A ∈ B)
23	12	adantlr 449	. . . 4 ⊢ (((φ ∧ y ∈ 𝐶) ∧ x = A) → (ψ ↔ χ))
24	22, 23	rspcedv 2637	. . 3 ⊢ ((φ ∧ y ∈ 𝐶) → (χ → ∃x ∈ B ψ))
25	24	rexlimdva 2411	. 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 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃wrex 2285

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537

This theorem is referenced by:  rexxfr2d  4147  rexxfr  4150