Theorem ralxfrd2 5384

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5380. (Contributed by Alexander van der Vekens, 25-Apr-2018.)

Hypotheses

Ref	Expression
ralxfrd2.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
ralxfrd2.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
ralxfrd2.3	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralxfrd2	⊢ (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 ralxfrd2

Step	Hyp	Ref	Expression
1		ralxfrd2.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)
2		ralxfrd2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3	2	3expa 1134	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rspcdv 3582	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	4	ralrimdva 3171	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐶 𝜒))
6		ralxfrd2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
7		r19.29 3134	. . . . 5 ⊢ ((∀𝑦 ∈ 𝐶 𝜒 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 (𝜒 ∧ 𝑥 = 𝐴))
8	2	ad4ant134 1191	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
9	8	exbiri 822	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → (𝜒 → 𝜓)))
10	9	impcomd 416	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → ((𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
11	10	rexlimdva 3172	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∃𝑦 ∈ 𝐶 (𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
12	7, 11	syl5 35	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑦 ∈ 𝐶 𝜒 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → 𝜓))
13	6, 12	mpan2d 706	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐶 𝜒 → 𝜓))
14	13	ralrimdva 3171	. 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 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096

This theorem is referenced by:  rexxfrd2  5385  ntrclsiso  44719  ntrclsk2  44720  ntrclskb  44721  ntrclsk3  44722  ntrclsk13  44723  ntrclsk4  44724  uptrlem1  49907  uptr2  49918