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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.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
| 2 | 3sstr3d.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 3 | 1, 2 | eqsstrrd 4019 | . 2 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
| 4 | 3sstr3d.3 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) | |
| 5 | 3, 4 | sseqtrd 4020 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ss 3968 |
| This theorem is referenced by: cnvtsr 18633 dprdss 20049 dprd2da 20062 dmdprdsplit2lem 20065 mplind 22094 txcmplem1 23649 setsmstopn 24490 tngtopn 24671 bcthlem2 25359 bcthlem4 25361 uniiccvol 25615 dyadmaxlem 25632 dvlip2 26034 dvne0 26050 shlej2 31380 gsumzresunsn 33059 pmtrcnel2 33110 cyc3co2 33160 ssdifidllem 33484 fedgmullem1 33680 hauseqcn 33897 bnd2lem 37798 heiborlem8 37825 dochord 41372 lclkrlem2p 41524 mapdsn 41643 hbtlem5 43140 oaabsb 43307 omabs2 43345 fvmptiunrelexplb0d 43697 fvmptiunrelexplb1d 43699 ovolval5lem3 46669 isclatd 48872 |
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