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Mirrors > Home > MPE Home > Th. List > 3sstr3d | Structured version Visualization version GIF version |
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.) |
Ref | Expression |
---|---|
3sstr3d.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
3sstr3d.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
3sstr3d.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
3sstr3d | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr3d.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 3sstr3d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
3 | 3sstr3d.3 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 2, 3 | sseq12d 3999 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷)) |
5 | 1, 4 | mpbid 234 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ⊆ wss 3935 |
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 2157 ax-12 2173 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 3942 df-ss 3951 |
This theorem is referenced by: cnvtsr 17826 dprdss 19145 dprd2da 19158 dmdprdsplit2lem 19161 mplind 20276 txcmplem1 22243 setsmstopn 23082 tngtopn 23253 bcthlem2 23922 bcthlem4 23924 uniiccvol 24175 dyadmaxlem 24192 dvlip2 24586 dvne0 24602 shlej2 29132 gsumzresunsn 30686 pmtrcnel2 30729 cyc3co2 30777 fedgmullem1 31020 hauseqcn 31133 bnd2lem 35063 heiborlem8 35090 dochord 38500 lclkrlem2p 38652 mapdsn 38771 hbtlem5 39721 fvmptiunrelexplb0d 40022 fvmptiunrelexplb1d 40024 |
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