Theorem rexxfr2d 4443

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

Hypotheses

Ref	Expression
ralxfr2d.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)
ralxfr2d.2	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))
ralxfr2d.3	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexxfr2d	⊢ (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)   𝑉(𝑥,𝑦)

Proof of Theorem rexxfr2d

Step	Hyp	Ref	Expression
1		ralxfr2d.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)
2		elisset 2740	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 𝑥 = 𝐴)
4		ralxfr2d.2	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))
5	4	biimprd 157	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
6		r19.23v 2575	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐶 (𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
7	5, 6	sylibr 133	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
8	7	r19.21bi 2554	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
9		eleq1 2229	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
10	8, 9	mpbidi 150	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
11	10	exlimdv 1807	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
12	3, 11	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
13	4	biimpa 294	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
14		ralxfr2d.3	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
15	12, 13, 14	rexxfrd 4441	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728

This theorem is referenced by:  rexrn  5622  rexima  5723  cnptopresti  12878  cnptoprest  12879  txrest  12916