Theorem sbcg 3797

Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3795. (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 3726	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		dfclel 2817	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}))
3		df-clab 2720	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
4		sbv 2100	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
5	3, 4	bitri 277	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6	5	anbi2i 630	. . . 4 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑦 = 𝐴 ∧ 𝜑))
7	6	exbii 1856	. . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
8	1, 2, 7	3bitrri 300	. 2 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑥]𝜑)
9		dfclel 2817	. . . 4 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉))
10	9	biimpi 218	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉))
11		simpr 486	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝜑) → 𝜑)
12	11	ax-gen 1803	. . . . 5 ⊢ ∀𝑦((𝑦 = 𝐴 ∧ 𝜑) → 𝜑)
13		19.23v 1950	. . . . . 6 ⊢ (∀𝑦((𝑦 = 𝐴 ∧ 𝜑) → 𝜑) ↔ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → 𝜑))
14	13	biimpi 218	. . . . 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 408	. . . . . . 7 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → (𝜑 → 𝑦 = 𝐴))
18	17	ancrd 557	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → (𝜑 → (𝑦 = 𝐴 ∧ 𝜑)))
19	18	eximi 1843	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → ∃𝑦(𝜑 → (𝑦 = 𝐴 ∧ 𝜑)))
20		19.37imv 1955	. . . . 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 214	. . 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 287	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   = wceq 1548  ∃wex 1787  [wsb 2074   ∈ wcel 2121  {cab 2719  [wsbc 3725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-clel 2816  df-sbc 3726

This theorem is referenced by:  sbcabel  3812  csbconstg  3852  2nreu  4375  csbuni  4871  csbxp  5722  sbcfung  6513  fmptsnd  7117  csbfrecsg  8228  opsbc2ie  32567  f1od2  32815  bnj89  34919  bnj525  34936  bnj1128  35187  csbrdgg  37706  csboprabg  37707  mptsnunlem  37715  topdifinffinlem  37724  relowlpssretop  37741  rdgeqoa  37747  csbfinxpg  37765  gm-sbtru  38488  sbfal  38489  cdlemk40  41424  cdlemkid3N  41440  cdlemkid4  41441  frege70  44392  frege77  44399  frege116  44438  frege118  44440  trsbc  44999  trsbcVD  45335  csbxpgVD  45352  csbunigVD  45356