Theorem ralbii2 1674

Description: Inference adding different restricted universal quantifiers to each side of an equivalence.

Hypothesis

Ref	Expression
ralbii2.1	⊢ ((x ∈ A → φ) ↔ (x ∈ B → ψ))

Assertion

Ref	Expression
ralbii2	⊢ (∀x ∈ A φ ↔ ∀x ∈ B ψ)

Proof of Theorem ralbii2

Step	Hyp	Ref	Expression
1		ralbii2.1	. . 3 ⊢ ((x ∈ A → φ) ↔ (x ∈ B → ψ))
2	1	albii 1001	. 2 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ B → ψ))
3		df-ral 1652	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
4		df-ral 1652	. 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ))
5	2, 3, 4	3bitr4 183	1 ⊢ (∀x ∈ A φ ↔ ∀x ∈ B ψ)

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 956   ∈ wcel 960  ∀wral 1648

This theorem is referenced by:  ralbiia 1676  zmin 6221  ralrp 6290  raluz2 6444  clm4 7080  h1det 9468

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1652

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