Theorem ralxfrALT 2906

Description: Transfer universal quantification from a variable x to another variable y contained in expression A.

Hypotheses

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

Assertion

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

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

Proof of Theorem ralxfrALT

Step	Hyp	Ref	Expression
1		eqid 1478	. 2 ⊢ V = V
2		ralxfr.1	. . . 4 ⊢ (y ∈ B → A ∈ B)
3	2	adantl 390	. . 3 ⊢ ((V = V ⋀ y ∈ B) → A ∈ B)
4		ralxfr.2	. . . 4 ⊢ (x ∈ B → ∃y ∈ B x = A)
5	4	adantl 390	. . 3 ⊢ ((V = V ⋀ x ∈ B) → ∃y ∈ B x = A)
6		ralxfr.3	. . . 4 ⊢ (x = A → (φ ↔ ψ))
7	6	adantl 390	. . 3 ⊢ ((V = V ⋀ x = A) → (φ ↔ ψ))
8	3, 5, 7	ralxfrd 2903	. 2 ⊢ (V = V → (∀x ∈ B φ ↔ ∀y ∈ B ψ))
9	1, 8	ax-mp 7	1 ⊢ (∀x ∈ B φ ↔ ∀y ∈ B ψ)

Syntax hints:   → wi 3   ↔ wb 146   = wceq 958   ∈ wcel 960  ∀wral 1648  ∃wrex 1649  Vcvv 1814

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815

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