Theorem rexsng 4608

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) (Proof shortened by AV, 7-Apr-2023.)

Hypothesis

Ref	Expression
ralsng.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexsng	⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rexsng

Step	Hyp	Ref	Expression
1		nfv 1911	. 2 ⊢ Ⅎ𝑥𝜓
2		ralsng.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	rexsngf 4604	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3497  df-sbc 3773  df-sn 4562

This theorem is referenced by:  rexsn  4614  rextpg  4629  iunxsng  5005  frirr  5527  frsn  5634  imasng  5946  scshwfzeqfzo  14182  dvdsprmpweqnn  16215  mnd1  17946  grp1  18200  elntg2  26765  1loopgrvd0  27280  1egrvtxdg0  27287  nfrgr2v  28045  1vwmgr  28049  elgrplsmsn  30939  ballotlemfc0  31745  ballotlemfcc  31746  bj-restsn  34367  elpaddat  36934  elpadd2at  36936  brfvidRP  40026  mnuunid  40606  iccelpart  43586  zlidlring  44192  lco0  44475