Theorem ralrimiv 2604

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 1576	. 2 ⊢ Ⅎ𝑥𝜑
2		ralrimiv.1	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	ralrimi 2603	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2202  ∀wral 2510

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515

This theorem is referenced by:  ralrimiva  2605  ralrimivw  2606  ralrimivv  2613  r19.27av  2668  rr19.3v  2945  rabssdv  3307  rzal  3592  trin  4197  class2seteq  4253  ralxfrALT  4564  ssorduni  4585  ordsucim  4598  onintonm  4615  issref  5119  funimaexglem  5413  resflem  5811  poxp  6396  rdgss  6548  dom2lem  6944  supisoti  7208  ordiso2  7233  updjud  7280  uzind  9590  zindd  9597  lbzbi  9849  icoshftf1o  10225  ccatrn  11185  ccatalpha  11189  maxabslemval  11768  xrmaxiflemval  11810  fisum0diag2  12007  alzdvds  12414  hashgcdeq  12811  ghmrn  13843  ghmpreima  13852  imasring  14076  01eq0ring  14202  islssmd  14372  tgcl  14787  distop  14808  neiuni  14884  cnpnei  14942  isxmetd  15070  fsumcncntop  15290  fsumdvdsmul  15714  uspgr2wlkeq  16215  clwwlkccatlem  16250  bj-nntrans2  16547  bj-inf2vnlem1  16565