Theorem rgen3 2712

Description: Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)

Hypothesis

Ref	Expression
rgen3.1	⊢ ((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ)

Assertion

Ref	Expression
rgen3	⊢ ∀x ∈ A ∀y ∈ B ∀z ∈ C φ

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

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

Proof of Theorem rgen3

Step	Hyp	Ref	Expression
1		rgen3.1	. . . 4 ⊢ ((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ)
2	1	3expa 1151	. . 3 ⊢ (((x ∈ A ∧ y ∈ B) ∧ z ∈ C) → φ)
3	2	ralrimiva 2698	. 2 ⊢ ((x ∈ A ∧ y ∈ B) → ∀z ∈ C φ)
4	3	rgen2 2711	1 ⊢ ∀x ∈ A ∀y ∈ B ∀z ∈ C φ

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   ∈ wcel 1710  ∀wral 2615

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-3an 936  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by: (None)