Theorem ralxfrd 5274

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.) (Proof shortened by JJ, 7-Aug-2021.)

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 714	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rspcdv 3563	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	4	ralrimdva 3154	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐶 𝜒))
6		ralxfrd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
7		r19.29 3216	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐶 𝜒 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 (𝜒 ∧ 𝑥 = 𝐴))
8	2	exbiri 810	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜒 → 𝜓)))
9	8	impcomd 415	. . . . . . 7 ⊢ (𝜑 → ((𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
10	9	rexlimdvw 3249	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 (𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
11	7, 10	syl5 34	. . . . 5 ⊢ (𝜑 → ((∀𝑦 ∈ 𝐶 𝜒 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → 𝜓))
12	11	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑦 ∈ 𝐶 𝜒 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → 𝜓))
13	6, 12	mpan2d 693	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐶 𝜒 → 𝜓))
14	13	ralrimdva 3154	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 𝜒 → ∀𝑥 ∈ 𝐵 𝜓))
15	5, 14	impbid 215	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112

This theorem is referenced by:  rexxfrd  5275  ralxfr2d  5276  ralxfr  5280  islindf4  20527  cmpfi  22013  rlimcnp  25551  ispisys2  31522  glbconN  36673  mapdordlem2  38933