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Theorem rexralbidv 2436
Description: Formula-building rule for restricted quantifiers (deduction form). (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 2412 . 2 (𝜑 → (∀𝑦𝐵 𝜓 ↔ ∀𝑦𝐵 𝜒))
32rexbidv 2413 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wral 2391  wrex 2392
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-ral 2396  df-rex 2397
This theorem is referenced by:  caucvgpr  7454  caucvgprpr  7484  caucvgsrlemgt1  7567  caucvgsrlemoffres  7572  axcaucvglemres  7671  cvg1nlemres  10708  rexfiuz  10712  resqrexlemgt0  10743  resqrexlemoverl  10744  resqrexlemglsq  10745  resqrexlemsqa  10747  resqrexlemex  10748  cau3lem  10837  caubnd2  10840  climi  11007  2clim  11021  ennnfonelemim  11843  lmcvg  12292  lmss  12321  txlm  12354  metcnpi  12590  metcnpi2  12591  elcncf  12635  cncfi  12640  limcimo  12709  cnplimclemr  12713  limccoap  12722
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