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

Theorem rspceov 5813
Description: A frequently used special case of rspc2ev 2804 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 5781 . . 3 (𝑥 = 𝐶 → (𝑥𝐹𝑦) = (𝐶𝐹𝑦))
21eqeq2d 2151 . 2 (𝑥 = 𝐶 → (𝑆 = (𝑥𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝑦)))
3 oveq2 5782 . . 3 (𝑦 = 𝐷 → (𝐶𝐹𝑦) = (𝐶𝐹𝐷))
43eqeq2d 2151 . 2 (𝑦 = 𝐷 → (𝑆 = (𝐶𝐹𝑦) ↔ 𝑆 = (𝐶𝐹𝐷)))
52, 4rspc2ev 2804 1 ((𝐶𝐴𝐷𝐵𝑆 = (𝐶𝐹𝐷)) → ∃𝑥𝐴𝑦𝐵 𝑆 = (𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 962   = wceq 1331  wcel 1480  wrex 2417  (class class class)co 5774
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  genpprecll  7334  genppreclu  7335  elz2  9134  znq  9428  qaddcl  9439  qmulcl  9441  qreccl  9446
  Copyright terms: Public domain W3C validator