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Theorem r19.29r 2570
Description: Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
Assertion
Ref Expression
r19.29r ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))

Proof of Theorem r19.29r
StepHypRef Expression
1 r19.29 2569 . 2 ((∀𝑥𝐴 𝜓 ∧ ∃𝑥𝐴 𝜑) → ∃𝑥𝐴 (𝜓𝜑))
2 ancom 264 . 2 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜓 ∧ ∃𝑥𝐴 𝜑))
3 ancom 264 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43rexbii 2442 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥𝐴 (𝜓𝜑))
51, 2, 43imtr4i 200 1 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wral 2416  wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-ral 2421  df-rex 2422
This theorem is referenced by:  r19.29af2  2572  lmss  12429  metcnp3  12694
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