Theorem hbral 1236

Description: Bound-variable hypothesis builder for restricted quantification.

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)

Distinct variable group(s):   x,y

Proof of Theorem hbral

Step	Hyp	Ref	Expression
1		hbral.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbral.2	. . . 4 ⊢ (φ → ∀xφ)
3	1, 2	hbim 702	. . 3 ⊢ ((y ∈ A → φ) → ∀x(y ∈ A → φ))
4	3	hbal 700	. 2 ⊢ (∀y(y ∈ A → φ) → ∀x∀y(y ∈ A → φ))
5		df-ral 1205	. 2 ⊢ (∀y ∈ A φ ↔ ∀y(y ∈ A → φ))
6	5	bial 695	. 2 ⊢ (∀x∀y ∈ A φ ↔ ∀x∀y(y ∈ A → φ))
7	4, 5, 6	3imtr4 192	1 ⊢ (∀y ∈ A φ → ∀x∀y ∈ A φ)

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  tfis 2245  ralxp 2456  f1fvf 2917  hbiso 2930  isotrALT 2936  hbrdg 2974  scottexs 3543  scott0s 3544  ac6lem 3575  hta 3619

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

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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