Theorem sbcg 3829

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3827. (Contributed by Alan Sare, 10-Nov-2012.) Reduce axiom usage. (Revised by GG, 12-Oct-2024.)

Assertion

Ref	Expression
sbcg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem sbcg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sbc 3757	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		dfclel 2805	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
3		df-clab 2709	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4		sbv 2089	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
5	3, 4	bitri 275	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6	5	anbi2i 623	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑦 = 𝐴 ∧ 𝜑))
7	6	exbii 1848	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
8	1, 2, 7	3bitrri 298	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑥]𝜑)
9		dfclel 2805	. . . 4 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉))
10	9	biimpi 216	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉))
11		simpr 484	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝜑) → 𝜑)
12	11	ax-gen 1795	. . . . 5 ⊢ ∀𝑦((𝑦 = 𝐴 ∧ 𝜑) → 𝜑)
13		19.23v 1942	. . . . . 6 ⊢ (∀𝑦((𝑦 = 𝐴 ∧ 𝜑) → 𝜑) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → 𝜑))
14	13	biimpi 216	. . . . 5 ⊢ (∀𝑦((𝑦 = 𝐴 ∧ 𝜑) → 𝜑) → (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → 𝜑))
15	12, 14	mp1i 13	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → 𝜑))
16		2a1 28	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑉 → (𝜑 → 𝑦 = 𝐴)))
17	16	imp 406	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → (𝜑 → 𝑦 = 𝐴))
18	17	ancrd 551	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → (𝜑 → (𝑦 = 𝐴 ∧ 𝜑)))
19	18	eximi 1835	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → ∃𝑦(𝜑 → (𝑦 = 𝐴 ∧ 𝜑)))
20		19.37imv 1947	. . . . 5 ⊢ (∃𝑦(𝜑 → (𝑦 = 𝐴 ∧ 𝜑)) → (𝜑 → ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
21	19, 20	syl 17	. . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → (𝜑 → ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)))
22	15, 21	impbid 212	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ 𝜑))
23	10, 22	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ 𝜑))
24	8, 23	bitr3id 285	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779  [wsb 2065   ∈ wcel 2109  {cab 2708  [wsbc 3756

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-clel 2804  df-sbc 3757

This theorem is referenced by:  sbcabel  3844  csbconstg  3884  2nreu  4410  csbuni  4903  csbxp  5741  sbcfung  6543  fmptsnd  7146  csbfrecsg  8266  opsbc2ie  32412  f1od2  32651  bnj89  34718  bnj525  34735  bnj1128  34987  csbrdgg  37324  csboprabg  37325  mptsnunlem  37333  topdifinffinlem  37342  relowlpssretop  37359  rdgeqoa  37365  csbfinxpg  37383  gm-sbtru  38107  sbfal  38108  cdlemk40  40918  cdlemkid3N  40934  cdlemkid4  40935  frege70  43929  frege77  43936  frege116  43975  frege118  43977  trsbc  44537  trsbcVD  44873  csbxpgVD  44890  csbunigVD  44894