Theorem hbralrimi 2937

Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 2940 and ralrimiv 2948. 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 1741	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 2901	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
5	3, 4	sylibr 223	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1473   ∈ wcel 1977  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-ral 2901

This theorem is referenced by:  ralrimi  2940  ralrimiv  2948  bnj1145  30315