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Theorem rspcedvdw 3614
Description: Version of rspcedvd 3613 where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on 𝜑, 𝑥. (Contributed by SN, 20-Aug-2024.)
Hypotheses
Ref Expression
rspcedvdw.s (𝑥 = 𝐴 → (𝜓𝜒))
rspcedvdw.1 (𝜑𝐴𝐵)
rspcedvdw.2 (𝜑𝜒)
Assertion
Ref Expression
rspcedvdw (𝜑 → ∃𝑥𝐵 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rspcedvdw
StepHypRef Expression
1 rspcedvdw.1 . 2 (𝜑𝐴𝐵)
2 rspcedvdw.2 . 2 (𝜑𝜒)
3 rspcedvdw.s . . 3 (𝑥 = 𝐴 → (𝜓𝜒))
43rspcev 3611 . 2 ((𝐴𝐵𝜒) → ∃𝑥𝐵 𝜓)
51, 2, 4syl2anc 582 1 (𝜑 → ∃𝑥𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wrex 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068
This theorem is referenced by:  opprring  20300  pzriprnglem7  21427  pzriprnglem13  21433  pzriprnglem14  21434  pzriprngALT  21435  remulscllem1  28256  remulscllem2  28257  fracerl  33025  fracfld  33027  idomsubr  33028  ghmqusnsglem1  33162  zringfrac  33285  ply1degltdimlem  33361  irredminply  33425  algextdeglem4  33429  algextdeglem8  33433  flt4lem2  42120  flt4lem7  42132  isuspgrim0lem  47265
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