MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reximddv2 Structured version   Visualization version   GIF version

Theorem reximddv2 3168
Description: Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
Hypotheses
Ref Expression
reximddv2.1 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)
reximddv2.2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
Assertion
Ref Expression
reximddv2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
Distinct variable groups:   𝑦,𝐴   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem reximddv2
StepHypRef Expression
1 reximddv2.1 . . . . 5 ((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)
21ex 397 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32reximdva 3165 . . 3 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 → ∃𝑦𝐵 𝜒))
43impr 442 . 2 ((𝜑 ∧ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)) → ∃𝑦𝐵 𝜒)
5 reximddv2.2 . 2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
64, 5reximddv 3166 1 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-ral 3066  df-rex 3067
This theorem is referenced by:  prmgaplem8  15969  cpmadugsumfi  20902  cpmidg2sum  20905  cayhamlem4  20913  ltgseg  25712  cgraswap  25933  cgracom  25935  cgratr  25936  dfcgra2  25942  xrofsup  29873  prmunb2  39036
  Copyright terms: Public domain W3C validator