Theorem ralxfrd 4447

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 474	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rspcdv 2837	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	4	ralrimdva 2550	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐶 𝜒))
6		ralxfrd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
7		r19.29 2607	. . . . 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 361	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜒) → 𝜓))
10	9	ancomsd 267	. . . . . . 7 ⊢ (𝜑 → ((𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
11	10	ad2antrr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → ((𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
12	11	rexlimdva 2587	. . . . 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 426	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐶 𝜒 → 𝜓))
15	14	ralrimdva 2550	. 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 1348   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732

This theorem is referenced by:  ralxfr2d  4449  ralxfr  4451