Theorem ralxfrd2 5374

Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5370. (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 1136	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rspcdv 3569	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	4	ralrimdva 3163	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐶 𝜒))
6		ralxfrd2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
7		r19.29 3126	. . . . 5 ⊢ ((∀𝑦 ∈ 𝐶 𝜒 ∧ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) → ∃𝑦 ∈ 𝐶 (𝜒 ∧ 𝑥 = 𝐴))
8	2	ad4ant134 1193	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
9	8	exbiri 823	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 → (𝜒 → 𝜓)))
10	9	impcomd 417	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐶) → ((𝜒 ∧ 𝑥 = 𝐴) → 𝜓))
11	10	rexlimdva 3164	. . . . 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 707	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐶 𝜒 → 𝜓))
14	13	ralrimdva 3163	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 𝜒 → ∀𝑥 ∈ 𝐵 𝜓))
15	5, 14	impbid 215	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088

This theorem is used by:  rexxfrd2  5375  ntrclsiso  45066  ntrclsk2  45067  ntrclskb  45068  ntrclsk3  45069  ntrclsk13  45070  ntrclsk4  45071  uptrlem1  50317  uptr2  50328