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Theorem r19.29r 2501
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 2500 . 2 ((∀𝑥𝐴 𝜓 ∧ ∃𝑥𝐴 𝜑) → ∃𝑥𝐴 (𝜓𝜑))
2 ancom 262 . 2 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜓 ∧ ∃𝑥𝐴 𝜑))
3 ancom 262 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43rexbii 2379 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥𝐴 (𝜓𝜑))
51, 2, 43imtr4i 199 1 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wral 2353  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-ral 2358  df-rex 2359
This theorem is referenced by:  r19.29af2  2502
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