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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 4041 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
| 2 | dfss2 3968 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
| 3 | 1, 2 | sylib 218 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∩ cin 3949 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-in 3957 df-ss 3967 | 
| This theorem is referenced by: bnj1321 35042 | 
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