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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  7684  caucvgprpr  7714  caucvgsrlemgt1  7797  caucvgsrlemoffres  7802  axcaucvglemres  7901  cvg1nlemres  10997  rexfiuz  11001  resqrexlemgt0  11032  resqrexlemoverl  11033  resqrexlemglsq  11034  resqrexlemsqa  11036  resqrexlemex  11037  cau3lem  11126  caubnd2  11129  climi  11298  2clim  11312  ennnfonelemim  12428  lmcvg  13878  lmss  13907  txlm  13940  metcnpi  14176  metcnpi2  14177  elcncf  14221  cncfi  14226  limcimo  14295  cnplimclemr  14299  limccoap  14308
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