Theorem ralsng 4655

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 2140, ax-12 2176. (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 3051	. . 3 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑))
2		velsn 4622	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	imbi1i 349	. . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑))
4	3	albii 1818	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))
5	1, 4	bitri 275	. 2 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))
6		elisset 2815	. . 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 1818	. . . . 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 1941	. . . . 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 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∀wral 3050  {csn 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-v 3465  df-sn 4607

This theorem is referenced by:  rexsng  4656  2ralsng  4658  ralsn  4661  ralprg  4676  raltpg  4678  ralunsn  4874  iinxsng  5068  frirr  5641  posn  5751  frsn  5753  f1ounsn  7274  f12dfv  7275  naddov2  8699  naddunif  8713  naddasslem1  8714  naddasslem2  8715  ranksnb  9849  mgm1  18640  sgrp1  18711  mnd1  18761  grp1  19034  cntzsnval  19311  abl1  19852  srgbinomlem4  20194  ring1  20275  mat1dimmul  22430  ufileu  23873  istrkg3ld  28405  1hevtxdg0  29451  wlkp1lem8  29626  wwlksnext  29841  wwlksext2clwwlk  30004  dfconngr1  30135  1conngr  30141  frgr1v  30218  lindssn  33341  lbslsat  33602  bj-raldifsn  37060  lindsadd  37579  poimirlem26  37612  poimirlem27  37613  poimirlem31  37617  cantnfresb  43299  safesnsupfilb  43393  cfsetsnfsetf1  47029  zlidlring  48108  linds0  48340  snlindsntor  48346  lmod1  48367