Theorem ralsng 4632

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 2184. (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 4596	. . . . 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 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 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 3051  {csn 4580

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3442  df-sn 4581

This theorem is referenced by:  rexsng  4633  2ralsng  4635  ralsn  4638  ralprg  4653  raltpg  4655  ralunsn  4850  iinxsng  5043  frirr  5600  posn  5710  frsn  5712  f1ounsn  7218  f12dfv  7219  naddov2  8607  naddunif  8621  naddasslem1  8622  naddasslem2  8623  ranksnb  9739  mgm1  18583  sgrp1  18654  mnd1  18704  grp1  18977  cntzsnval  19253  abl1  19795  srgbinomlem4  20164  ring1  20245  mat1dimmul  22420  ufileu  23863  sltssnb  27765  eqcuts3  27800  bdayn0p1  28365  istrkg3ld  28533  1hevtxdg0  29579  wlkp1lem8  29752  wwlksnext  29966  wwlksext2clwwlk  30132  dfconngr1  30263  1conngr  30269  frgr1v  30346  lindssn  33459  lbslsat  33773  bj-raldifsn  37301  lindsadd  37810  poimirlem26  37843  poimirlem27  37844  poimirlem31  37848  cantnfresb  43562  safesnsupfilb  43655  cfsetsnfsetf1  47301  zlidlring  48476  linds0  48707  snlindsntor  48713  lmod1  48734