Theorem ralsng 4613

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 7-Apr-2023.)

Hypothesis

Ref	Expression
ralsng.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralsng	⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ralsng

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
2		ralsng.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	ralsngf 4611	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {csn 4567

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 2793

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-sbc 3773  df-sn 4568

This theorem is referenced by:  2ralsng  4616  ralsn  4619  raltpg  4634  ralunsn  4824  iinxsng  5010  frirr  5532  posn  5637  frsn  5639  f12dfv  7030  ranksnb  9256  mgm1  17868  sgrp1  17910  mnd1  17952  grp1  18206  cntzsnval  18454  abl1  18986  srgbinomlem4  19293  ring1  19352  mat1dimmul  21085  ufileu  22527  istrkg3ld  26247  1hevtxdg0  27287  wlkp1lem8  27462  wwlksnext  27671  wwlksext2clwwlk  27836  dfconngr1  27967  1conngr  27973  frgr1v  28050  lindssn  30939  lbslsat  31014  bj-raldifsn  34395  lindsadd  34900  poimirlem26  34933  poimirlem27  34934  poimirlem31  34938  zlidlring  44219  linds0  44540  snlindsntor  44546  lmod1  44567