Theorem ralrimia 39835

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 397	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	1, 3	ralrimi 3106	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 382  Ⅎwnf 1856   ∈ wcel 2145  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-nf 1858  df-ral 3066

This theorem is referenced by:  ralimda  39846  funimaeq  39976  ralrnmpt3  39989  rnmptssbi  39992  fconst7  39999  infleinf2  40154  unb2ltle  40155  uzublem  40170  climinf3  40463  limsupequzlem  40469  limsupre3uzlem  40482  climisp  40493  climrescn  40495  climxrrelem  40496  climxrre  40497  climxlim2lem  40586  meaiuninc3v  41215