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Theorem srhmsubcALTVlem1 41884
 Description: Lemma 1 for srhmsubcALTV 41887. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
srhmsubcALTV.s 𝑟𝑆 𝑟 ∈ Ring
srhmsubcALTV.c 𝐶 = (𝑈𝑆)
Assertion
Ref Expression
srhmsubcALTVlem1 (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
Distinct variable groups:   𝑆,𝑟   𝑋,𝑟
Allowed substitution hints:   𝐶(𝑟)   𝑈(𝑟)

Proof of Theorem srhmsubcALTVlem1
StepHypRef Expression
1 eleq1 2676 . . . 4 (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring))
2 srhmsubcALTV.s . . . 4 𝑟𝑆 𝑟 ∈ Ring
31, 2vtoclri 3256 . . 3 (𝑋𝑆𝑋 ∈ Ring)
43anim2i 591 . 2 ((𝑋𝑈𝑋𝑆) → (𝑋𝑈𝑋 ∈ Ring))
5 srhmsubcALTV.c . . 3 𝐶 = (𝑈𝑆)
65elin2 3763 . 2 (𝑋𝐶 ↔ (𝑋𝑈𝑋𝑆))
7 elin 3758 . 2 (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋𝑈𝑋 ∈ Ring))
84, 6, 73imtr4i 280 1 (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539  Ringcrg 18370 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-in 3547 This theorem is referenced by:  srhmsubcALTVlem2  41885
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