Theorem sbcg 3802

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3800. (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 3730	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		dfclel 2813	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
3		df-clab 2716	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4		sbv 2094	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
5	3, 4	bitri 275	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6	5	anbi2i 624	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑦 = 𝐴 ∧ 𝜑))
7	6	exbii 1850	. . 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 2813	. . . 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 1797	. . . . 5 ⊢ ∀𝑦((𝑦 = 𝐴 ∧ 𝜑) → 𝜑)
13		19.23v 1944	. . . . . 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 1837	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → ∃𝑦(𝜑 → (𝑦 = 𝐴 ∧ 𝜑)))
20		19.37imv 1949	. . . . 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 1540   = wceq 1542  ∃wex 1781  [wsb 2068   ∈ wcel 2114  {cab 2715  [wsbc 3729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-clel 2812  df-sbc 3730

This theorem is referenced by:  sbcabel  3817  csbconstg  3857  2nreu  4385  csbuni  4881  csbxp  5723  sbcfung  6514  fmptsnd  7115  csbfrecsg  8225  opsbc2ie  32565  f1od2  32812  bnj89  34885  bnj525  34902  bnj1128  35153  csbrdgg  37656  csboprabg  37657  mptsnunlem  37665  topdifinffinlem  37674  relowlpssretop  37691  rdgeqoa  37697  csbfinxpg  37715  gm-sbtru  38438  sbfal  38439  cdlemk40  41374  cdlemkid3N  41390  cdlemkid4  41391  frege70  44375  frege77  44382  frege116  44421  frege118  44423  trsbc  44982  trsbcVD  45318  csbxpgVD  45335  csbunigVD  45339