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Theorem rexralbidv 2503
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 2477 . 2 (𝜑 → (∀𝑦𝐵 𝜓 ↔ ∀𝑦𝐵 𝜒))
32rexbidv 2478 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wral 2455  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461
This theorem is referenced by:  caucvgpr  7681  caucvgprpr  7711  caucvgsrlemgt1  7794  caucvgsrlemoffres  7799  axcaucvglemres  7898  cvg1nlemres  10994  rexfiuz  10998  resqrexlemgt0  11029  resqrexlemoverl  11030  resqrexlemglsq  11031  resqrexlemsqa  11033  resqrexlemex  11034  cau3lem  11123  caubnd2  11126  climi  11295  2clim  11309  ennnfonelemim  12425  lmcvg  13720  lmss  13749  txlm  13782  metcnpi  14018  metcnpi2  14019  elcncf  14063  cncfi  14068  limcimo  14137  cnplimclemr  14141  limccoap  14150
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