![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3877 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
2 | sseq2 3878 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) | |
3 | 1, 2 | sylan9bb 502 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ⊆ wss 3824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-in 3831 df-ss 3838 |
This theorem is referenced by: sseq12i 3882 sorpsscmpl 7277 funcnvuni 7450 fun11iun 7457 sornom 9496 axdc3lem2 9670 ipole 17639 ipodrsima 17646 metsscmetcld 23637 funpsstri 32561 brredunds 35339 ismrcd2 38725 ismrc 38727 |
Copyright terms: Public domain | W3C validator |