Theorem 2ralbidva 2530

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 1552	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1552	. 2 ⊢ Ⅎ𝑦𝜑
3		2ralbidva.1	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))
4	1, 2, 3	2ralbida 2529	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2178  ∀wral 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-4 1534  ax-17 1550

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-ral 2491

This theorem is referenced by:  soinxp  4763  isotr  5908  fnmpoovd  6324  sgrppropd  13360  mndpropd  13387  mhmpropd  13413  cmnpropd  13746  rngpropd  13832  ringpropd  13915  lmodprop2d  14225  lsspropdg  14308  ismet2  14941  txmetcn  15106