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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 464 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))) |
4 | eqss 3112 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | 3, 4 | syl6bbr 197 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ⊆ wss 3071 |
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-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: (None) |
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