Theorem hbral 2663

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)

Hypotheses

Ref	Expression
hbral.1	⊢ (y ∈ A → ∀x y ∈ A)
hbral.2	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
hbral	⊢ (∀y ∈ A φ → ∀x∀y ∈ A φ)

Proof of Theorem hbral

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 ⊢ (∀y ∈ A φ ↔ ∀y(y ∈ A → φ))
2		hbral.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
3		hbral.2	. . . 4 ⊢ (φ → ∀xφ)
4	2, 3	hbim 1817	. . 3 ⊢ ((y ∈ A → φ) → ∀x(y ∈ A → φ))
5	4	hbal 1736	. 2 ⊢ (∀y(y ∈ A → φ) → ∀x∀y(y ∈ A → φ))
6	1, 5	hbxfrbi 1568	1 ⊢ (∀y ∈ A φ → ∀x∀y ∈ A φ)

Syntax hints:   → wi 4  ∀wal 1540   ∈ 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-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by: (None)