Theorem hbra1 3299

Description: The setvar 𝑥 is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Proof shortened by Wolf Lammen, 7-Dec-2019.)

Assertion

Ref	Expression
hbra1	⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑)

Proof of Theorem hbra1

Step	Hyp	Ref	Expression
1		nfra1 3286	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
2	1	nf5ri 2231	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1812  df-nf 1816  df-ral 3077

This theorem is used by:  bnj1095  35462  bnj1309  35702  mpobi123f  39231  hbra2VD  45978  tratrbVD  45979  ssralv2VD  45984