Theorem rexsng 4610

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) Avoid ax-10 2137, ax-12 2171. (Revised by Gino Giotto, 30-Sep-2024.)

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		ralsng.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	notbid 318	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	ralsng 4609	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓))
4		dfrex2 3170	. . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
5		bicom1 220	. . . 4 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑))
6	5	con1bid 356	. . 3 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ 𝜓))
7	4, 6	bitrid 282	. 2 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
8	3, 7	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-sn 4562

This theorem is referenced by:  rexsn  4618  rextpg  4635  iunxsng  5019  frirr  5566  frsn  5674  imasng  5991  scshwfzeqfzo  14539  dvdsprmpweqnn  16586  mnd1  18426  grp1  18682  elntg2  27353  1loopgrvd0  27871  1egrvtxdg0  27878  nfrgr2v  28636  1vwmgr  28640  elgrplsmsn  31578  grplsmid  31592  ballotlemfc0  32459  ballotlemfcc  32460  addscllem1  34131  bj-restsn  35253  elpaddat  37818  elpadd2at  37820  brfvidRP  41296  mnuunid  41895  iccelpart  44885  zlidlring  45486  lco0  45768