Theorem hbralrimi 3143

Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 3256 and ralrimiv 3144. Its main advantage over these two is its minimal references to axioms. The proof is extracted from NM's previous work. (Contributed by Wolf Lammen, 4-Dec-2019.)

Hypotheses

Ref	Expression
hbralrimi.1	⊢ (𝜑 → ∀𝑥𝜑)
hbralrimi.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))

Assertion

Ref	Expression
hbralrimi	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Proof of Theorem hbralrimi

Step	Hyp	Ref	Expression
1		hbralrimi.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		hbralrimi.2	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	alrimih 1823	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 3061	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
5	3, 4	sylibr 234	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1537   ∈ wcel 2107  ∀wral 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808

This theorem depends on definitions:  df-bi 207  df-ral 3061

This theorem is referenced by:  ralrimiv  3144  ralrimi  3256  bnj1145  35008