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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 473 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))) |
4 | eqss 3172 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | 3, 4 | bitr4di 198 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ⊆ wss 3131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3137 df-ss 3144 |
This theorem is referenced by: (None) |
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