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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 3928 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷) |
5 | 1, 4 | sylib 221 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⊆ wss 3863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3407 df-in 3870 df-ss 3880 |
This theorem is referenced by: complss 4058 uniintsn 4895 fpwwe2lem12 10253 hmeocls 22662 hmeontr 22663 usgrumgruspgr 27268 chsscon3i 29539 pjss1coi 30241 mdslmd2i 30408 satffunlem2lem2 33078 ssbnd 35681 bnd2lem 35684 trclubgNEW 40900 nzss 41606 |
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