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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 13114 | . 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | df-bj-ind 13114 | . . 3 ⊢ (Ind 𝐵 ↔ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵)) | |
3 | eleq2 2201 | . . . . 5 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵)) | |
4 | 3 | bicomd 140 | . . . 4 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐵 ↔ ∅ ∈ 𝐴)) |
5 | eleq2 2201 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐵)) | |
6 | 5 | raleqbi1dv 2632 | . . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵)) |
7 | 6 | bicomd 140 | . . . 4 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) |
8 | 4, 7 | anbi12d 464 | . . 3 ⊢ (𝐴 = 𝐵 → ((∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))) |
9 | 2, 8 | syl5rbb 192 | . 2 ⊢ (𝐴 = 𝐵 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) ↔ Ind 𝐵)) |
10 | 1, 9 | syl5bb 191 | 1 ⊢ (𝐴 = 𝐵 → (Ind 𝐴 ↔ Ind 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2414 ∅c0 3358 suc csuc 4282 Ind wind 13113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-bj-ind 13114 |
This theorem is referenced by: bj-omind 13121 bj-omssind 13122 bj-ssom 13123 bj-om 13124 bj-2inf 13125 |
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