Theorem rexsng 4657

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) Avoid ax-10 2142, ax-12 2178. (Revised by GG, 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 4656	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓))
4		dfrex2 3064	. . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
5		bicom1 221	. . . 4 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ 𝜓 ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑))
6	5	con1bid 355	. . 3 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓) → (¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ 𝜓))
7	4, 6	bitrid 283	. 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 206   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  {csn 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-v 3466  df-sn 4607

This theorem is referenced by:  rexsn  4663  rextpg  4680  iunxsng  5071  frirr  5635  frsn  5747  imasng  6076  naddunif  8710  scshwfzeqfzo  14850  dvdsprmpweqnn  16910  mnd1  18762  grp1  19035  pzriprnglem3  21449  pzriprnglem10  21456  psdmul  22109  cutmax  27899  cutmin  27900  halfcut  28390  elntg2  28969  1loopgrvd0  29489  1egrvtxdg0  29496  nfrgr2v  30258  1vwmgr  30262  elgrplsmsn  33410  grplsmid  33424  ballotlemfc0  34530  ballotlemfcc  34531  bj-restsn  37105  elrnressn  38296  elpaddat  39828  elpadd2at  39830  brfvidRP  43679  mnuunid  44268  iccelpart  47414  zlidlring  48176  lco0  48370