Theorem ralsng 4619

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2147, ax-12 2185. (Revised by GG, 30-Sep-2024.)

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		df-ral 3052	. . 3 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑))
2		velsn 4583	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	imbi1i 349	. . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑))
4	3	albii 1821	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))
5	1, 4	bitri 275	. 2 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))
6		elisset 2818	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
7		ralsng.1	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
8	7	pm5.74i 271	. . . . . 6 ⊢ ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓))
9	8	albii 1821	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓))
10	9	a1i 11	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜓)))
11		19.23v 1944	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))
12	11	a1i 11	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
13		pm5.5 361	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴 → 𝜓) ↔ 𝜓))
14	10, 12, 13	3bitrd 305	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
15	6, 14	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
16	5, 15	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3051  {csn 4567

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3431  df-sn 4568

This theorem is referenced by:  rexsng  4620  2ralsng  4622  ralsn  4625  ralprg  4640  raltpg  4642  ralunsn  4837  iinxsng  5030  frirr  5607  posn  5717  frsn  5719  f1ounsn  7227  f12dfv  7228  naddov2  8615  naddunif  8629  naddasslem1  8630  naddasslem2  8631  ranksnb  9751  mgm1  18626  sgrp1  18697  mnd1  18747  grp1  19023  cntzsnval  19299  abl1  19841  srgbinomlem4  20210  ring1  20291  mat1dimmul  22441  ufileu  23884  sltssnb  27761  eqcuts3  27796  bdayn0p1  28361  istrkg3ld  28529  1hevtxdg0  29574  wlkp1lem8  29747  wwlksnext  29961  wwlksext2clwwlk  30127  dfconngr1  30258  1conngr  30264  frgr1v  30341  lindssn  33438  lbslsat  33760  bj-raldifsn  37412  lindsadd  37934  poimirlem26  37967  poimirlem27  37968  poimirlem31  37972  cantnfresb  43752  safesnsupfilb  43845  cfsetsnfsetf1  47507  zlidlring  48710  linds0  48941  snlindsntor  48947  lmod1  48968