Theorem rexsng 4608

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) Avoid ax-10 2152, ax-12 2189. (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 319	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	ralsng 4607	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓))
4		dfrex2 3066	. . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
5		bicom1 222	. . . 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 284	. 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 207   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  {csn 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-v 3433  df-sn 4556

This theorem is referenced by:  rexsn  4614  rextpg  4631  iunxsng  5019  frirr  5594  frsn  5706  imasng  6036  naddunif  8619  snecg  8714  scshwfzeqfzo  14779  dvdsprmpweqnn  16847  mnd1  18738  grp1  19014  pzriprnglem3  21458  pzriprnglem10  21465  psdmul  22154  cutmax  27944  cutmin  27945  halfcut  28468  elntg2  29072  1loopgrvd0  29591  1egrvtxdg0  29598  nfrgr2v  30360  1vwmgr  30364  elgrplsmsn  33473  grplsmid  33487  ballotlemfc0  34677  ballotlemfcc  34678  bj-restsn  37440  elrnressn  38647  elpaddat  40296  elpadd2at  40298  brfvidRP  44132  mnuunid  44721  iccelpart  47908  zlidlring  48725  lco0  48918