ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexralbidv GIF version

Theorem rexralbidv 2400
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 2376 . 2 (𝜑 → (∀𝑦𝐵 𝜓 ↔ ∀𝑦𝐵 𝜒))
32rexbidv 2377 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wral 2355  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-ral 2360  df-rex 2361
This theorem is referenced by:  caucvgpr  7188  caucvgprpr  7218  caucvgsrlemgt1  7287  caucvgsrlemoffres  7292  axcaucvglemres  7381  cvg1nlemres  10317  rexfiuz  10321  resqrexlemgt0  10352  resqrexlemoverl  10353  resqrexlemglsq  10354  resqrexlemsqa  10356  resqrexlemex  10357  cau3lem  10446  caubnd2  10449  climi  10573  2clim  10587
  Copyright terms: Public domain W3C validator