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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.2 | . . 3 ⊢ 𝐴 = 𝐶 | |
| 2 | 3sstr3g.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 3 | 1, 2 | eqsstrrid 4023 | . 2 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
| 4 | 3sstr3g.3 | . 2 ⊢ 𝐵 = 𝐷 | |
| 5 | 3, 4 | sseqtrdi 4024 | 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: complss 4151 uniintsn 4985 fpwwe2lem12 10682 hmeocls 23776 hmeontr 23777 usgrumgruspgr 29199 chsscon3i 31480 pjss1coi 32182 mdslmd2i 32349 satffunlem2lem2 35411 ssbnd 37795 bnd2lem 37798 trclubgNEW 43631 nzss 44336 |
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