Theorem rexsng 4616

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 1915	. 2 ⊢ Ⅎ𝑥𝜓
2		ralsng.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	rexsngf 4612	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  {csn 4569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rex 3146  df-v 3498  df-sbc 3775  df-sn 4570

This theorem is referenced by:  rexsn  4622  rextpg  4637  iunxsng  5014  frirr  5534  frsn  5641  imasng  5953  scshwfzeqfzo  14190  dvdsprmpweqnn  16223  mnd1  17954  grp1  18208  elntg2  26773  1loopgrvd0  27288  1egrvtxdg0  27295  nfrgr2v  28053  1vwmgr  28057  elgrplsmsn  30946  ballotlemfc0  31752  ballotlemfcc  31753  bj-restsn  34375  elpaddat  36942  elpadd2at  36944  brfvidRP  40040  mnuunid  40620  iccelpart  43600  zlidlring  44206  lco0  44489