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Theorem reximddv2 3213
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 414 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32reximdva 3169 . . 3 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 → ∃𝑦𝐵 𝜒))
43impr 456 . 2 ((𝜑 ∧ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)) → ∃𝑦𝐵 𝜒)
5 reximddv2.2 . 2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
64, 5reximddv 3172 1 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wrex 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-rex 3072
This theorem is referenced by:  r19.29vva  3214  r19.29vvaOLD  3215  prmgaplem8  16991  cpmadugsumfi  22379  cpmidg2sum  22382  cayhamlem4  22390  ltgseg  27847  cgraswap  28071  cgracom  28073  cgratr  28074  flatcgra  28075  dfcgra2  28081  xrofsup  31980  prmunb2  43070
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