Theorem rgen3 1265

Description: Generalization rule for restricted quantification.

Hypothesis

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

Assertion

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

Distinct variable group(s):   y,z,A   x,z

Proof of Theorem rgen3

Step	Hyp	Ref	Expression
1		rgen3.1	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → φ)
2	1	3exp 611	. . . 4 ⊢ (x ∈ A → (y ∈ A → (z ∈ A → φ)))
3	2	imp 277	. . 3 ⊢ ((x ∈ A ∧ y ∈ A) → (z ∈ A → φ))
4	3	r19.21aiv 1259	. 2 ⊢ ((x ∈ A ∧ y ∈ A) → ∀z ∈ A φ)
5	4	rgen2 1248	1 ⊢ ∀x ∈ A ∀y ∈ A ∀z ∈ A φ

Syntax hints:   → wi 2   ∧ wa 196   ∧ w3a 581   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  itlso 2151  zorn2 3612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-cleq 1097  df-clel 1099  df-ral 1205

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