Theorem bi2ral 1225

Description: Inference adding 2 restricted universal quantifiers to both sides of an equivalence.

Hypothesis

Ref	Expression
biral.1	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
bi2ral	⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x ∈ A ∀y ∈ B ψ)

Proof of Theorem bi2ral

Step	Hyp	Ref	Expression
1		biral.1	. . 3 ⊢ (φ ↔ ψ)
2	1	biral 1223	. 2 ⊢ (∀y ∈ B φ ↔ ∀y ∈ B ψ)
3	2	biral 1223	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x ∈ A ∀y ∈ B ψ)

Syntax hints:   ↔ wb 127  ∀wral 1201

This theorem is referenced by:  reu4 1340  fununi 2705  zorn2 3612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

metamath.org