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

Theorem sbcreug 2919
Description: Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
sbcreug (𝐴𝑉 → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcreug
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2843 . 2 (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑[𝐴 / 𝑥]∃!𝑦𝐵 𝜑))
2 dfsbcq2 2843 . . 3 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32reubidv 2550 . 2 (𝑧 = 𝐴 → (∃!𝑦𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
4 nfcv 2228 . . . 4 𝑥𝐵
5 nfs1v 1863 . . . 4 𝑥[𝑧 / 𝑥]𝜑
64, 5nfreuxy 2541 . . 3 𝑥∃!𝑦𝐵 [𝑧 / 𝑥]𝜑
7 sbequ12 1701 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
87reubidv 2550 . . 3 (𝑥 = 𝑧 → (∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑))
96, 8sbie 1721 . 2 ([𝑧 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝑧 / 𝑥]𝜑)
101, 3, 9vtoclbg 2680 1 (𝐴𝑉 → ([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wcel 1438  [wsb 1692  ∃!wreu 2361  [wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-reu 2366  df-v 2621  df-sbc 2841
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator