Theorem hbral 3146

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 3069	. 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 2296	. . 3 ⊢ ((𝑦 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑦 ∈ 𝐴 → 𝜑))
5	4	hbal 2167	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜑) → ∀𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
6	1, 5	hbxfrbi 1827	1 ⊢ (∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥∀𝑦 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∀wal 1537   ∈ wcel 2106  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787  df-ral 3069

This theorem is referenced by:  nfralw  3151  tratrbVD  42481