Theorem 2ralbii 2417

Description: Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)

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 2415	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)
3	2	ralbii 2415	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)

Syntax hints:   ↔ wb 104  ∀wral 2390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-ral 2395

This theorem is referenced by:  rmo4f  2851  ordsoexmid  4437  cnvsom  5040  fununi  5149  tpossym  6127  isbasis2g  12055