Theorem ralrimia 3234

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 412	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	1, 3	ralrimi 3233	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2184

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-ral 3050

This theorem is referenced by:  ralimdaa  3236  iineq2d  4947  funcnvmpt  6938  fompt  7059  rnmptssd  7065  vieta  33712  ss2iundf  44074  ismnushort  44716  modelaxreplem3  45395  ssrabdf  45533  ss2rabdf  45568  iunssdf  45574  dmmptdff  45640  axccd  45646  dmmptdf2  45650  rnmptbd2lem  45665  rnmptssdf  45671  rnmptbdlem  45672  ralrnmpt3  45676  rnmptssbi  45677  fconst7  45681  fmptdff  45688  rnmptssdff  45692  infleinf2  45830  unb2ltle  45831  uzublem  45846  cvgcaule  45907  climinf3  46132  limsupequzlem  46138  limsupre3uzlem  46151  climisp  46162  climrescn  46164  climxrrelem  46165  climxrre  46166  climxlim2lem  46261  dvnprodlem1  46362  saliunclf  46738  meaiuninc3v  46900  preimageiingt  47136  preimaleiinlt  47137  fsupdm  47258  finfdm  47262  iinfssc  49520  iinfsubc  49521