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Theorem rspceov 7192
Description: A frequently used special case of rspc2ev 3632 for operation values. (Contributed by NM, 21-Mar-2007.)
Assertion
Ref Expression
rspceov ((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑦,𝐷   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐷(𝑥)

Proof of Theorem rspceov
StepHypRef Expression
1 oveq1 7152 . . 3 (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦))
21eqeq2d 2829 . 2 (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦)))
3 oveq2 7153 . . 3 (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷))
43eqeq2d 2829 . 2 (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷)))
52, 4rspc2ev 3632 1 ((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  iunfictbso  9528  genpprecl  10411  elz2  11987  zaddcl  12010  znq  12340  qaddcl  12352  qmulcl  12354  qreccl  12356  xpsff1o  16828  mndpfo  17922  gafo  18364  lsmelvalix  18695  lsmelvalmi  18706  evthicc2  23988  i1fadd  24223  i1fmul  24224  2clwwlk2clwwlk  28056  isgrpoi  28202  shscli  29021  shsva  29024  shunssi  29072  pjpjhth  29129  spanunsni  29283  pjjsi  29404  ofrn2  30315  pstmfval  31035  ismblfin  34814  itg2addnc  34827  blbnd  34946  isgrpda  35114  sbgoldbalt  43823
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