MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexsngOLD Structured version   Visualization version   GIF version

Theorem rexsngOLD 4676
Description: Obsolete version of rexsng 4672 as of 30-Sep-2024. (Contributed by NM, 29-Jan-2012.) (Proof shortened by AV, 7-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ralsngOLD.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexsngOLD (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rexsngOLD
StepHypRef Expression
1 nfv 1909 . 2 𝑥𝜓
2 ralsngOLD.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2rexsngf 4668 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wrex 3060  {csn 4622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rex 3061  df-v 3465  df-sbc 3769  df-sn 4623
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator