Theorem ralrimia 3237

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 3236	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052

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 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-ral 3053

This theorem is referenced by:  funcnvmpt  6953  fompt  7074  rnmptssd  7080  vieta  33763  ss2iundf  44044  ismnushort  44686  modelaxreplem3  45365  ssrabdf  45503  ss2rabdf  45538  iunssdf  45544  dmmptdff  45610  axccd  45616  dmmptdf2  45620  rnmptbd2lem  45635  rnmptssdf  45641  rnmptbdlem  45642  ralrnmpt3  45646  rnmptssbi  45647  fconst7  45651  fmptdff  45658  rnmptssdff  45662  infleinf2  45801  unb2ltle  45802  uzublem  45817  cvgcaule  45878  climinf3  46103  limsupequzlem  46109  limsupre3uzlem  46122  climisp  46133  climrescn  46135  climxrrelem  46136  climxrre  46137  climxlim2lem  46232  dvnprodlem1  46333  saliunclf  46709  meaiuninc3v  46871  preimageiingt  47107  preimaleiinlt  47108  fsupdm  47229  finfdm  47233  iinfssc  49445  iinfsubc  49446