Theorem ralrimiv 2566

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 1539	. 2 ⊢ Ⅎ𝑥𝜑
2		ralrimiv.1	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	ralrimi 2565	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2164  ∀wral 2472

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477

This theorem is referenced by:  ralrimiva  2567  ralrimivw  2568  ralrimivv  2575  r19.27av  2629  rr19.3v  2900  rabssdv  3260  rzal  3545  trin  4138  class2seteq  4193  ralxfrALT  4499  ssorduni  4520  ordsucim  4533  onintonm  4550  issref  5049  funimaexglem  5338  resflem  5723  poxp  6287  rdgss  6438  dom2lem  6828  supisoti  7071  ordiso2  7096  updjud  7143  uzind  9431  zindd  9438  lbzbi  9684  icoshftf1o  10060  maxabslemval  11355  xrmaxiflemval  11396  fisum0diag2  11593  alzdvds  11999  hashgcdeq  12380  ghmrn  13330  ghmpreima  13339  imasring  13563  01eq0ring  13688  islssmd  13858  tgcl  14243  distop  14264  neiuni  14340  cnpnei  14398  isxmetd  14526  fsumcncntop  14746  bj-nntrans2  15514  bj-inf2vnlem1  15532