Theorem 2ralbii 1672

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

Hypothesis

Ref	Expression
ralbii.1	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
2ralbii	⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x ∈ A ∀y ∈ B ψ)

Proof of Theorem 2ralbii

Step	Hyp	Ref	Expression
1		ralbii.1	. . 3 ⊢ (φ ↔ ψ)
2	1	ralbii 1670	. 2 ⊢ (∀y ∈ B φ ↔ ∀y ∈ B ψ)
3	2	ralbii 1670	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x ∈ A ∀y ∈ B ψ)

Syntax hints:   ↔ wb 146  ∀wral 1648

This theorem is referenced by:  cnvso 3529  fununi 3569  zorn 4807  isbasis2g 7611  dfadj2 9807  adjval2t 9810  cnlnadjeu 10005  adjbdlnt 10011

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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