Theorem ralrimivva 2706

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)

Hypothesis

Ref	Expression
ralrimivva.1	⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → ψ)

Assertion

Ref	Expression
ralrimivva	⊢ (φ → ∀x ∈ A ∀y ∈ B ψ)

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

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

Proof of Theorem ralrimivva

Step	Hyp	Ref	Expression
1		ralrimivva.1	. . 3 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → ψ)
2	1	ex 423	. 2 ⊢ (φ → ((x ∈ A ∧ y ∈ B) → ψ))
3	2	ralrimivv 2705	1 ⊢ (φ → ∀x ∈ A ∀y ∈ B ψ)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619

This theorem is referenced by:  nnpw1ex  4484  isocnv  5491  isotr  5495  f1oiso  5499  caovcld  5622  caovcomg  5624  antird  5928  iserd  5942  ncspw1eu  6159