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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 4025 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷) |
5 | 1, 4 | sylib 218 | 1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ⊆ wss 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-ss 3979 |
This theorem is referenced by: complss 4160 uniintsn 4989 fpwwe2lem12 10679 hmeocls 23791 hmeontr 23792 usgrumgruspgr 29213 chsscon3i 31489 pjss1coi 32191 mdslmd2i 32358 satffunlem2lem2 35390 ssbnd 37774 bnd2lem 37777 trclubgNEW 43607 nzss 44312 |
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