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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-sseq | GIF version |
Description: If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
bj-sseq.1 | ⊢ (𝜑 → (𝜓 ↔ 𝐴 ⊆ 𝐵)) |
bj-sseq.2 | ⊢ (𝜑 → (𝜒 ↔ 𝐵 ⊆ 𝐴)) |
Ref | Expression |
---|---|
bj-sseq | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-sseq.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝐴 ⊆ 𝐵)) | |
2 | bj-sseq.2 | . . 3 ⊢ (𝜑 → (𝜒 ↔ 𝐵 ⊆ 𝐴)) | |
3 | 1, 2 | anbi12d 442 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))) |
4 | eqss 2960 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | 3, 4 | syl6bbr 187 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ⊆ wss 2917 |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 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-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: (None) |
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