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Theorem reximddv2 3014
 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 450 . . . 4 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32reximdva 3012 . . 3 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 → ∃𝑦𝐵 𝜒))
43impr 648 . 2 ((𝜑 ∧ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜓)) → ∃𝑦𝐵 𝜒)
5 reximddv2.2 . 2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
64, 5reximddv 3013 1 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1987  ∃wrex 2908 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-ral 2912  df-rex 2913 This theorem is referenced by:  prmgaplem8  15697  cpmadugsumfi  20614  cpmidg2sum  20617  cayhamlem4  20625  ltgseg  25408  cgraswap  25629  cgracom  25631  cgratr  25632  dfcgra2  25638  xrofsup  29400  prmunb2  38027
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