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Mirrors > Home > MPE Home > Th. List > 3sstr3i | Structured version Visualization version GIF version |
Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
3sstr3.1 | ⊢ 𝐴 ⊆ 𝐵 |
3sstr3.2 | ⊢ 𝐴 = 𝐶 |
3sstr3.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3sstr3i | ⊢ 𝐶 ⊆ 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr3.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | 3sstr3.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
3 | 3sstr3.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | sseq12i 3780 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷) |
5 | 1, 4 | mpbi 220 | 1 ⊢ 𝐶 ⊆ 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ⊆ wss 3723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-in 3730 df-ss 3737 |
This theorem is referenced by: odf1o2 18195 leordtval2 21237 uniiccvol 23568 ballotlem2 30890 cotrcltrcl 38543 |
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