Theorem ralrimia 40060

Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
ralrimia.1	⊢ Ⅎ𝑥𝜑
ralrimia.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)

Assertion

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

Proof of Theorem ralrimia

Step	Hyp	Ref	Expression
1		ralrimia.1	. 2 ⊢ Ⅎ𝑥𝜑
2		ralrimia.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	ex 402	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	1, 3	ralrimi 3136	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 385  Ⅎwnf 1879   ∈ wcel 2157  ∀wral 3087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880  df-ral 3092

This theorem is referenced by:  ralimda  40071  funimaeq  40196  ralrnmpt3  40209  rnmptssbi  40212  fconst7  40219  infleinf2  40372  unb2ltle  40373  uzublem  40388  climinf3  40680  limsupequzlem  40686  limsupre3uzlem  40699  climisp  40710  climrescn  40712  climxrrelem  40713  climxrre  40714  climxlim2lem  40803  meaiuninc3v  41432