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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 13296 | . 2 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | df-bj-ind 13296 | . . 3 ⊢ (Ind 𝐵 ↔ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 suc 𝑥 ∈ 𝐵)) | |
3 | eleq2 2204 | . . . . 5 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵)) | |
4 | 3 | bicomd 140 | . . . 4 ⊢ (𝐴 = 𝐵 → (∅ ∈ 𝐵 ↔ ∅ ∈ 𝐴)) |
5 | eleq2 2204 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝑥 ∈ 𝐵)) | |
6 | 5 | raleqbi1dv 2637 | . . . . 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 465 | . . 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 1332 ∈ wcel 1481 ∀wral 2417 ∅c0 3368 suc csuc 4295 Ind wind 13295 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-bj-ind 13296 |
This theorem is referenced by: bj-omind 13303 bj-omssind 13304 bj-ssom 13305 bj-om 13306 bj-2inf 13307 |
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