Theorem ralsng 4642

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 2142, ax-12 2178. (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 3046	. . 3 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑))
2		velsn 4608	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	imbi1i 349	. . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑))
4	3	albii 1819	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))
5	1, 4	bitri 275	. 2 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))
6		elisset 2811	. . 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 1819	. . . . 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 1942	. . . . 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 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  {csn 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-sn 4593

This theorem is referenced by:  rexsng  4643  2ralsng  4645  ralsn  4648  ralprg  4663  raltpg  4665  ralunsn  4861  iinxsng  5055  frirr  5617  posn  5727  frsn  5729  f1ounsn  7250  f12dfv  7251  naddov2  8646  naddunif  8660  naddasslem1  8661  naddasslem2  8662  ranksnb  9787  mgm1  18592  sgrp1  18663  mnd1  18713  grp1  18986  cntzsnval  19263  abl1  19803  srgbinomlem4  20145  ring1  20226  mat1dimmul  22370  ufileu  23813  bdayn0p1  28265  istrkg3ld  28395  1hevtxdg0  29440  wlkp1lem8  29615  wwlksnext  29830  wwlksext2clwwlk  29993  dfconngr1  30124  1conngr  30130  frgr1v  30207  lindssn  33356  lbslsat  33619  bj-raldifsn  37095  lindsadd  37614  poimirlem26  37647  poimirlem27  37648  poimirlem31  37652  cantnfresb  43320  safesnsupfilb  43414  cfsetsnfsetf1  47064  zlidlring  48226  linds0  48458  snlindsntor  48464  lmod1  48485