Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  elabg2 GIF version

Theorem elabg2 10746
 Description: One implication of elabg 2740. (Contributed by BJ, 21-Nov-2019.)
Hypothesis
Ref Expression
elabg2.1 (𝑥 = 𝐴 → (𝜓𝜑))
Assertion
Ref Expression
elabg2 (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elabg2
StepHypRef Expression
1 nfcv 2220 . 2 𝑥𝐴
2 nfv 1462 . 2 𝑥𝜓
3 elabg2.1 . 2 (𝑥 = 𝐴 → (𝜓𝜑))
41, 2, 3elabgf2 10741 1 (𝐴𝑉 → (𝜓𝐴 ∈ {𝑥𝜑}))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1285   ∈ wcel 1434  {cab 2068 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator