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Theorem r19.29r 3055
Description: Restricted quantifier version of 19.29r 1790; variation of r19.29 3054. (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 3054 . 2 ((∀𝑥𝐴 𝜓 ∧ ∃𝑥𝐴 𝜑) → ∃𝑥𝐴 (𝜓𝜑))
2 ancom 465 . 2 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜓 ∧ ∃𝑥𝐴 𝜑))
3 ancom 465 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43rexbii 3023 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥𝐴 (𝜓𝜑))
51, 2, 43imtr4i 280 1 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) → ∃𝑥𝐴 (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wral 2896  wrex 2897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902
This theorem is referenced by:  r19.29imd  3056  2reu5  3383  rlimuni  14129  rlimno1  14232  neindisj2  20737  lmss  20912  fclsbas  21635  isfcf  21648  ucnima  21895  metcnp3  22155  cfilucfil  22174  bndth  22565  ellimc3  23449  lmxrge0  29326  gsumesum  29448  esumcst  29452  esumfsup  29459  voliune  29619  volfiniune  29620  bnj517  30209  cover2  32678  prmunb2  37532
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