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Mirrors > Home > MPE Home > Th. List > sseq12 | Structured version Visualization version GIF version |
Description: Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.) |
Ref | Expression |
---|---|
sseq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3992 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
2 | sseq2 3993 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) | |
3 | 1, 2 | sylan9bb 512 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ⊆ wss 3936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 |
This theorem is referenced by: sseq12i 3997 sorpsscmpl 7454 funcnvuni 7630 fiunlem 7637 sornom 9693 axdc3lem2 9867 ipole 17762 ipodrsima 17769 metsscmetcld 23912 funpsstri 33003 brredunds 35855 ismrcd2 39289 ismrc 39291 |
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