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Theorem r19.29d2r 3326
Description: Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.)
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)
3 r19.29 3248 . . 3 ((∀𝑥𝐴𝑦𝐵 𝜓 ∧ ∃𝑥𝐴𝑦𝐵 𝜒) → ∃𝑥𝐴 (∀𝑦𝐵 𝜓 ∧ ∃𝑦𝐵 𝜒))
41, 2, 3syl2anc 587 . 2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 𝜓 ∧ ∃𝑦𝐵 𝜒))
5 r19.29 3248 . . 3 ((∀𝑦𝐵 𝜓 ∧ ∃𝑦𝐵 𝜒) → ∃𝑦𝐵 (𝜓𝜒))
65reximi 3237 . 2 (∃𝑥𝐴 (∀𝑦𝐵 𝜓 ∧ ∃𝑦𝐵 𝜒) → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
74, 6syl 17 1 (𝜑 → ∃𝑥𝐴𝑦𝐵 (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wral 3133  wrex 3134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3138  df-rex 3139
This theorem is referenced by:  r19.29vvaOLD  3328  ucnima  22896  tgisline  26430  r19.29ffa  30253  xrofsup  30513  icoreresf  34741  sn-negex12  39511
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