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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 9386 | . 2 ⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) | |
2 | df-bj-ind 9386 | . . 3 ⊢ (Ind B ↔ (∅ ∈ B ∧ ∀x ∈ B suc x ∈ B)) | |
3 | eleq2 2098 | . . . . 5 ⊢ (A = B → (∅ ∈ A ↔ ∅ ∈ B)) | |
4 | 3 | bicomd 129 | . . . 4 ⊢ (A = B → (∅ ∈ B ↔ ∅ ∈ A)) |
5 | eleq2 2098 | . . . . . 6 ⊢ (A = B → (suc x ∈ A ↔ suc x ∈ B)) | |
6 | 5 | raleqbi1dv 2507 | . . . . 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 1242 ∈ wcel 1390 ∀wral 2300 ∅c0 3218 suc csuc 4068 Ind wind 9385 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-bj-ind 9386 |
This theorem is referenced by: bj-omind 9393 bj-omssind 9394 bj-ssom 9395 bj-om 9396 bj-2inf 9397 peano5setOLD 9400 |
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