Theorem ralxfrd 4321

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)

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
ralxfrd	⊢ (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 ralxfrd

Step	Hyp	Ref	Expression
1		ralxfrd.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
2		ralxfrd.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3	2	adantlr 464	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rspcdv 2747	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	4	ralrimdva 2471	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐶 𝜒))
6		ralxfrd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
7		r19.29 2528	. . . . 5 ⊢ ((∀𝑦 ∈ 𝐶 𝜒 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 (𝜒 ∧ 𝑥 = 𝐴))
8	2	biimprd 157	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
9	8	expimpd 358	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜒) → 𝜓))
10	9	ancomsd 267	. . . . . . 7 ⊢ (𝜑 → ((𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
11	10	ad2antrr 475	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → ((𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
12	11	rexlimdva 2508	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∃𝑦 ∈ 𝐶 (𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
13	7, 12	syl5 32	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑦 ∈ 𝐶 𝜒 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → 𝜓))
14	6, 13	mpan2d 422	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐶 𝜒 → 𝜓))
15	14	ralrimdva 2471	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 𝜒 → ∀𝑥 ∈ 𝐵 𝜓))
16	5, 15	impbid 128	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1299   ∈ wcel 1448  ∀wral 2375  ∃wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643

This theorem is referenced by:  ralxfr2d  4323  ralxfr  4325