Theorem ralxfrALT 4165

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. This proof does not use ralxfrd 4160. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
ralxfrALT	⊢ (∀x ∈ B φ ↔ ∀y ∈ 𝐶 ψ)

Distinct variable groups:   ψ,x   φ,y   x,A   x,y,B   x,𝐶

Allowed substitution hints:   φ(x)   ψ(y)   A(y)   𝐶(y)

Proof of Theorem ralxfrALT

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . . 5 ⊢ (y ∈ 𝐶 → A ∈ B)
2		ralxfr.3	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
3	2	rspcv 2646	. . . . 5 ⊢ (A ∈ B → (∀x ∈ B φ → ψ))
4	1, 3	syl 14	. . . 4 ⊢ (y ∈ 𝐶 → (∀x ∈ B φ → ψ))
5	4	com12 27	. . 3 ⊢ (∀x ∈ B φ → (y ∈ 𝐶 → ψ))
6	5	ralrimiv 2385	. 2 ⊢ (∀x ∈ B φ → ∀y ∈ 𝐶 ψ)
7		ralxfr.2	. . . 4 ⊢ (x ∈ B → ∃y ∈ 𝐶 x = A)
8		nfra1 2349	. . . . 5 ⊢ Ⅎy∀y ∈ 𝐶 ψ
9		nfv 1418	. . . . 5 ⊢ Ⅎyφ
10		rsp 2363	. . . . . 6 ⊢ (∀y ∈ 𝐶 ψ → (y ∈ 𝐶 → ψ))
11	2	biimprcd 149	. . . . . 6 ⊢ (ψ → (x = A → φ))
12	10, 11	syl6 29	. . . . 5 ⊢ (∀y ∈ 𝐶 ψ → (y ∈ 𝐶 → (x = A → φ)))
13	8, 9, 12	rexlimd 2424	. . . 4 ⊢ (∀y ∈ 𝐶 ψ → (∃y ∈ 𝐶 x = A → φ))
14	7, 13	syl5 28	. . 3 ⊢ (∀y ∈ 𝐶 ψ → (x ∈ B → φ))
15	14	ralrimiv 2385	. 2 ⊢ (∀y ∈ 𝐶 ψ → ∀x ∈ B φ)
16	6, 15	impbii 117	1 ⊢ (∀x ∈ B φ ↔ ∀y ∈ 𝐶 ψ)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553

This theorem is referenced by: (None)