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Mirrors > Home > ILE Home > Th. List > eqsstrd | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
eqsstrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqsstrd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
eqsstrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
2 | eqsstrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | sseq1d 3199 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
4 | 1, 3 | mpbird 167 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ⊆ wss 3144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-in 3150 df-ss 3157 |
This theorem is referenced by: eqsstrrd 3207 eqsstrdi 3222 tfisi 4604 funresdfunsnss 5740 suppssof1 6125 pw2f1odclem 6863 phplem4dom 6891 fival 7000 fiuni 7008 cardonle 7217 exmidfodomrlemim 7231 frecuzrdgtclt 10454 4sqlem19 12444 ennnfonelemkh 12466 ennnfonelemf1 12472 strfvssn 12537 setscom 12555 imasaddfnlemg 12794 imasaddflemg 12796 reldvdsrsrg 13459 tgrest 14146 resttopon 14148 rest0 14156 lmtopcnp 14227 metequiv2 14473 xmettx 14487 ellimc3apf 14606 dvfvalap 14627 dvcjbr 14649 dvcj 14650 dvfre 14651 nnsf 15233 |
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