Theorem ralrimiv 2602

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)

Hypothesis

Ref	Expression
ralrimiv.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))

Assertion

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

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem ralrimiv

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜑
2		ralrimiv.1	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	ralrimi 2601	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2200  ∀wral 2508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-4 1556  ax-17 1572

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513

This theorem is referenced by:  ralrimiva  2603  ralrimivw  2604  ralrimivv  2611  r19.27av  2666  rr19.3v  2942  rabssdv  3304  rzal  3589  trin  4192  class2seteq  4247  ralxfrALT  4558  ssorduni  4579  ordsucim  4592  onintonm  4609  issref  5111  funimaexglem  5404  resflem  5801  poxp  6384  rdgss  6535  dom2lem  6931  supisoti  7188  ordiso2  7213  updjud  7260  uzind  9569  zindd  9576  lbzbi  9823  icoshftf1o  10199  ccatrn  11157  ccatalpha  11161  maxabslemval  11735  xrmaxiflemval  11777  fisum0diag2  11974  alzdvds  12381  hashgcdeq  12778  ghmrn  13810  ghmpreima  13819  imasring  14043  01eq0ring  14169  islssmd  14339  tgcl  14754  distop  14775  neiuni  14851  cnpnei  14909  isxmetd  15037  fsumcncntop  15257  fsumdvdsmul  15681  uspgr2wlkeq  16111  clwwlkccatlem  16143  bj-nntrans2  16398  bj-inf2vnlem1  16416