Theorem sbcg 3812

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3810. (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 3740	. . 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 2090	. . . . . 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 1849	. . 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 1796	. . . . 5 ⊢ ∀𝑦((𝑦 = 𝐴 ∧ 𝜑) → 𝜑)
13		19.23v 1943	. . . . . 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 1836	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → ∃𝑦(𝜑 → (𝑦 = 𝐴 ∧ 𝜑)))
20		19.37imv 1948	. . . . 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 1539   = wceq 1541  ∃wex 1780  [wsb 2066   ∈ wcel 2110  {cab 2708  [wsbc 3739

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 2112

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-clel 2804  df-sbc 3740

This theorem is referenced by:  sbcabel  3827  csbconstg  3867  2nreu  4392  csbuni  4886  csbxp  5714  sbcfung  6501  fmptsnd  7098  csbfrecsg  8209  opsbc2ie  32445  f1od2  32692  bnj89  34723  bnj525  34740  bnj1128  34992  csbrdgg  37342  csboprabg  37343  mptsnunlem  37351  topdifinffinlem  37360  relowlpssretop  37377  rdgeqoa  37383  csbfinxpg  37401  gm-sbtru  38125  sbfal  38126  cdlemk40  40935  cdlemkid3N  40951  cdlemkid4  40952  frege70  43945  frege77  43952  frege116  43991  frege118  43993  trsbc  44552  trsbcVD  44888  csbxpgVD  44905  csbunigVD  44909