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Theorem rexralbidv 2367
Description: Formula-building rule for restricted quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
Hypothesis
Ref Expression
2ralbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
rexralbidv (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 rexralbidv
StepHypRef Expression
1 2ralbidv.1 . . 3 (𝜑 → (𝜓𝜒))
21ralbidv 2343 . 2 (𝜑 → (∀𝑦𝐵 𝜓 ↔ ∀𝑦𝐵 𝜒))
32rexbidv 2344 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328  df-rex 2329
This theorem is referenced by:  caucvgpr  6837  caucvgprpr  6867  caucvgsrlemgt1  6936  caucvgsrlemoffres  6941  axcaucvglemres  7030  cvg1nlemres  9811  rexfiuz  9815  resqrexlemgt0  9846  resqrexlemoverl  9847  resqrexlemglsq  9848  resqrexlemsqa  9850  resqrexlemex  9851  cau3lem  9940  caubnd2  9943  climi  10038  2clim  10052
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