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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1322 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1322 | ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 3620 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | df-ss 3554 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
3 | 1, 2 | sylib 207 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∩ cin 3539 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 |
This theorem is referenced by: bnj1321 30349 |
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