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Mirrors > Home > MPE Home > Th. List > 3sstr3g | 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 |
---|---|
3sstr3g.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
3sstr3g.2 | ⊢ 𝐴 = 𝐶 |
3sstr3g.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3sstr3g | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3sstr3g.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 3sstr3g.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
3 | 3sstr3g.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | sseq12i 4007 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷) |
5 | 1, 4 | sylib 217 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ⊆ wss 3943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-ss 3960 |
This theorem is referenced by: complss 4141 uniintsn 4984 fpwwe2lem12 10636 hmeocls 23622 hmeontr 23623 usgrumgruspgr 28943 chsscon3i 31218 pjss1coi 31920 mdslmd2i 32087 satffunlem2lem2 34924 ssbnd 37168 bnd2lem 37171 trclubgNEW 42927 nzss 43634 |
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