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Mirrors > Home > MPE Home > Th. List > sbcg | Structured version Visualization version GIF version |
Description: Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3868. (Contributed by Alan Sare, 10-Nov-2012.) Reduce axiom usage. (Revised by GG, 12-Oct-2024.) |
Ref | Expression |
---|---|
sbcg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3792 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
2 | dfclel 2815 | . . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ {𝑥 ∣ 𝜑})) | |
3 | df-clab 2713 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
4 | sbv 2086 | . . . . . 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 1845 | . . 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 2815 | . . . 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 1792 | . . . . 5 ⊢ ∀𝑦((𝑦 = 𝐴 ∧ 𝜑) → 𝜑) |
13 | 19.23v 1940 | . . . . . 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 1832 | . . . . 5 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑉) → ∃𝑦(𝜑 → (𝑦 = 𝐴 ∧ 𝜑))) |
20 | 19.37imv 1945 | . . . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1776 [wsb 2062 ∈ wcel 2106 {cab 2712 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: sbcabel 3887 csbconstg 3927 2nreu 4450 csbuni 4941 csbxp 5788 sbcfung 6592 fmptsnd 7189 csbfrecsg 8308 opsbc2ie 32504 f1od2 32739 bnj89 34714 bnj525 34731 bnj1128 34983 csbrdgg 37312 csboprabg 37313 mptsnunlem 37321 topdifinffinlem 37330 relowlpssretop 37347 rdgeqoa 37353 csbfinxpg 37371 gm-sbtru 38093 sbfal 38094 cdlemk40 40900 cdlemkid3N 40916 cdlemkid4 40917 frege70 43923 frege77 43930 frege116 43969 frege118 43971 trsbc 44538 trsbcVD 44875 csbxpgVD 44892 csbunigVD 44896 |
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