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Theorem 2rexbidva 3216
Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
Hypothesis
Ref Expression
2ralbidva.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
2rexbidva (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Distinct variable groups:   𝑥,𝑦,𝜑   𝑦,𝐴
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem 2rexbidva
StepHypRef Expression
1 2ralbidva.1 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
21anassrs 467 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32rexbidva 3175 . 2 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 ↔ ∃𝑦𝐵 𝜒))
43rexbidva 3175 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2105  wrex 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-rex 3070
This theorem is referenced by:  2reu4lem  4525  wrdl3s3  14918  bezoutlem2  16487  bezoutlem4  16489  vdwmc2  16917  lsmcom2  19565  lsmass  19579  lsmcomx  19766  lsmspsn  20840  hausdiag  23370  imasf1oxms  24219  mulsval  27805  mulscom  27835  addsdi  27850  mulsasslem3  27860  istrkg2ld  27979  iscgra  28328  axeuclid  28489  elwwlks2  29488  elwspths2spth  29489  fusgr2wsp2nb  29855  shscom  30840  lsmssass  32787  sategoelfvb  34709  3dim0  38632  islpln5  38710  islvol5  38754  isline2  38949  isline3  38951  paddcom  38988  cdlemg2cex  39766  prprspr2  46485  pgrpgt2nabl  47131  elbigolo1  47331
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