Theorem ralsng 4627

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 2146, ax-12 2182. (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 3049	. . 3 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑))
2		velsn 4591	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	imbi1i 349	. . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑))
4	3	albii 1820	. . 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 1820	. . . . 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 1943	. . . . 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 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3048  {csn 4575

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-v 3439  df-sn 4576

This theorem is referenced by:  rexsng  4628  2ralsng  4630  ralsn  4633  ralprg  4648  raltpg  4650  ralunsn  4845  iinxsng  5038  frirr  5595  posn  5705  frsn  5707  f1ounsn  7212  f12dfv  7213  naddov2  8600  naddunif  8614  naddasslem1  8615  naddasslem2  8616  ranksnb  9727  mgm1  18568  sgrp1  18639  mnd1  18689  grp1  18962  cntzsnval  19238  abl1  19780  srgbinomlem4  20149  ring1  20230  mat1dimmul  22392  ufileu  23835  ssltsnb  27733  eqscut3  27766  bdayn0p1  28295  istrkg3ld  28440  1hevtxdg0  29486  wlkp1lem8  29659  wwlksnext  29873  wwlksext2clwwlk  30039  dfconngr1  30170  1conngr  30176  frgr1v  30253  lindssn  33350  lbslsat  33650  bj-raldifsn  37165  lindsadd  37673  poimirlem26  37706  poimirlem27  37707  poimirlem31  37711  cantnfresb  43441  safesnsupfilb  43535  cfsetsnfsetf1  47183  zlidlring  48358  linds0  48590  snlindsntor  48596  lmod1  48617