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

Theorem reximddv2 3276
 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 415 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32reximdva 3272 . . 3 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 → ∃𝑦𝐵 𝜒))
43impr 457 . 2 ((𝜑 ∧ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)) → ∃𝑦𝐵 𝜒)
5 reximddv2.2 . 2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
64, 5reximddv 3273 1 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2107  ∃wrex 3137 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 1904 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-ral 3141  df-rex 3142 This theorem is referenced by:  r19.29vva  3334  prmgaplem8  16386  cpmadugsumfi  21477  cpmidg2sum  21480  cayhamlem4  21488  ltgseg  26374  cgraswap  26598  cgracom  26600  cgratr  26601  flatcgra  26602  dfcgra2  26608  xrofsup  30484  prmunb2  40628
 Copyright terms: Public domain W3C validator