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Theorem r19.29d2r 3130
Description: Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
Hypotheses
Ref Expression
r19.29d2r.1 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
r19.29d2r.2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
Assertion
Ref Expression
r19.29d2r (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))

Proof of Theorem r19.29d2r
StepHypRef Expression
1 r19.29d2r.1 . . 3 (𝜑 → ∀𝑥𝐴𝑦𝐵 𝜓)
2 r19.29d2r.2 . . 3 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜒)
31, 2jca 510 . 2 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ∧ ∃𝑥𝐴𝑦𝐵 𝜒))
4 2r19.29 3129 . 2 ((∀𝑥𝐴𝑦𝐵 𝜓 ∧ ∃𝑥𝐴𝑦𝐵 𝜒) → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
53, 4syl 17 1 (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wral 3051  wrex 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-ral 3052  df-rex 3061
This theorem is referenced by:  ucnima  24277  tgisline  28554  r19.29ffa  32401  xrofsup  32671  icoreresf  37059  sn-negex12  42196
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