Theorem 2ralbidva 2572

Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)

Hypothesis

Ref	Expression
2ralbidva.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
2ralbidva	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Distinct variable groups:   𝑥,𝑦,𝜑   𝑦,𝐴

Allowed substitution hints:   𝜓(𝑥, 𝑦)   𝜒(𝑥, 𝑦)   𝐴(𝑥)   𝐵(𝑥, 𝑦)

Proof of Theorem 2ralbidva

Step	Hyp	Ref	Expression
1		nfv 1581	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1581	. 2 ⊢ Ⅎ𝑦𝜑
3		2ralbidva.1	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))
4	1, 2, 3	2ralbida 2571	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2209  ∀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-nf 1514  df-ral 2533

This theorem is used by:  soinxp  4845  isotr  6022  fnmpoovd  6451  sgrppropd  13728  mndpropd  13753  mhmpropd  13773  cmnpropd  14098  rngpropd  14254  ringpropd  14343  lmodprop2d  14685  lsspropdg  14768  assapropd  15014  ismet2  15455  txmetcn  15620