Theorem hbral 3307

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 3078	. 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 2333	. . 3 ⊢ ((𝑦 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑦 ∈ 𝐴 → 𝜑))
5	4	hbal 2204	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜑) → ∀𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
6	1, 5	hbxfrbi 1858	1 ⊢ (∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥∀𝑦 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   ∈ wcel 2145  ∀wral 3077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817  df-ral 3078

This theorem is used by:  nfralw  3310  tratrbVD  45828