Theorem ralrimivvva 2708

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypothesis

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

Assertion

Ref	Expression
ralrimivvva	⊢ (φ → ∀x ∈ A ∀y ∈ B ∀z ∈ C ψ)

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

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

Proof of Theorem ralrimivvva

Step	Hyp	Ref	Expression
1		ralrimivvva.1	. . . . . 6 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B ∧ z ∈ C)) → ψ)
2	1	3exp2 1169	. . . . 5 ⊢ (φ → (x ∈ A → (y ∈ B → (z ∈ C → ψ))))
3	2	imp41 576	. . . 4 ⊢ ((((φ ∧ x ∈ A) ∧ y ∈ B) ∧ z ∈ C) → ψ)
4	3	ralrimiva 2698	. . 3 ⊢ (((φ ∧ x ∈ A) ∧ y ∈ B) → ∀z ∈ C ψ)
5	4	ralrimiva 2698	. 2 ⊢ ((φ ∧ x ∈ A) → ∀y ∈ B ∀z ∈ C ψ)
6	5	ralrimiva 2698	1 ⊢ (φ → ∀x ∈ A ∀y ∈ B ∀z ∈ C ψ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ w3a 934   ∈ wcel 1710  ∀wral 2615

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is used by:  caovassg  5627  caovdig  5633  caovdirg  5634