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Theorem 2rexbidv 2463
 Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
Hypothesis
Ref Expression
2ralbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
2rexbidv (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem 2rexbidv
StepHypRef Expression
1 2ralbidv.1 . . 3 (𝜑 → (𝜓𝜒))
21rexbidv 2439 . 2 (𝜑 → (∃𝑦𝐵 𝜓 ↔ ∃𝑦𝐵 𝜒))
32rexbidv 2439 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∃wrex 2418 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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-rex 2423 This theorem is referenced by:  f1oiso  5735  elrnmpog  5891  elrnmpo  5892  ralrnmpo  5893  rexrnmpo  5894  ovelrn  5927  eroveu  6528  genipv  7342  genpelxp  7344  genpelvl  7345  genpelvu  7346  axcnre  7714  apreap  8374  apreim  8390  aprcl  8433  bezoutlemnewy  11721  bezoutlema  11724  bezoutlemb  11725  txuni2  12465  txbas  12467  txdis1cn  12487
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