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Theorem rexbidva 2373
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 9-Mar-1997.)
Hypothesis
Ref Expression
ralbidva.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rexbidva (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rexbidva
StepHypRef Expression
1 nfv 1464 . 2 𝑥𝜑
2 ralbidva.1 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
31, 2rexbida 2371 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1436  wrex 2356
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-rex 2361
This theorem is referenced by:  2rexbiia  2390  2rexbidva  2397  rexeqbidva  2573  dfimafn  5316  funimass4  5318  fconstfvm  5476  fliftel  5533  fliftf  5539  f1oiso  5566  releldm2  5912  frecabcl  6118  qsinxp  6320  qliftel  6324  supisolem  6647  genpassl  7027  genpassu  7028  addcomprg  7081  mulcomprg  7083  1idprl  7093  1idpru  7094  archrecnq  7166  archrecpr  7167  caucvgprprlemexbt  7209  caucvgprprlemexb  7210  archsr  7271  cnegexlem3  7603  cnegex2  7605  recexre  7996  rerecclap  8136  creur  8354  creui  8355  nndiv  8397  arch  8603  nnrecl  8604  expnlbnd  9975  fimaxq  10132  clim2  10566  clim2c  10567  clim0c  10569  climabs0  10590  climrecvg1n  10629  sumeq2  10640  nndivides  10685  alzdvds  10737  oddm1even  10757  oddnn02np1  10762  oddge22np1  10763  evennn02n  10764  evennn2n  10765  divalgb  10807  modremain  10811
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