Theorem 2ralbii 2558

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 2556	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)
3	2	ralbii 2556	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   ↔ wb 105  ∀wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-ral 2533

This theorem is used by:  rmo4f  3024  ordsoexmid  4709  cnvsom  5331  fununi  5449  tpossym  6547  axpre-suploc  8269  issubm  13781  isbasis2g  15148  ivthdich  15756