Theorem rexsng 4677

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 317	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	ralsng 4676	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ ¬ 𝜓))
4		dfrex2 3073	. . 3 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ¬ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
5		bicom1 220	. . . 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 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 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {csn 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-v 3476  df-sn 4628

This theorem is referenced by:  rexsn  4685  rextpg  4702  iunxsng  5092  frirr  5652  frsn  5761  imasng  6079  naddunif  8688  scshwfzeqfzo  14773  dvdsprmpweqnn  16814  mnd1  18663  grp1  18926  elntg2  28232  1loopgrvd0  28750  1egrvtxdg0  28757  nfrgr2v  29514  1vwmgr  29518  elgrplsmsn  32488  grplsmid  32502  ballotlemfc0  33479  ballotlemfcc  33480  bj-restsn  35951  elrnressn  37129  elpaddat  38663  elpadd2at  38665  brfvidRP  42424  mnuunid  43021  iccelpart  46087  zlidlring  46779  lco0  47061