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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-indeq | GIF version |
Description: Equality property for Ind. (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
bj-indeq | ⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-ind 10051 | . 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | df-bj-ind 10051 | . . 3 ⊢ (Ind 𝐵 ↔ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵)) | |
3 | eleq2 2101 | . . . . 5 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵)) | |
4 | 3 | bicomd 129 | . . . 4 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐵 ↔ ∅ ∈ 𝐴)) |
5 | eleq2 2101 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐵)) | |
6 | 5 | raleqbi1dv 2513 | . . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵)) |
7 | 6 | bicomd 129 | . . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) |
8 | 4, 7 | anbi12d 442 | . . 3 ⊢ (𝐴 = 𝐵 → ((∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))) |
9 | 2, 8 | syl5rbb 182 | . 2 ⊢ (𝐴 = 𝐵 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) ↔ Ind 𝐵)) |
10 | 1, 9 | syl5bb 181 | 1 ⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ∅c0 3224 suc csuc 4102 Ind wind 10050 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-bj-ind 10051 |
This theorem is referenced by: bj-omind 10058 bj-omssind 10059 bj-ssom 10060 bj-om 10061 bj-2inf 10062 peano5setOLD 10065 |
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