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Mirrors > Home > ILE Home > Th. List > sseqtrrdi | GIF version |
Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
sseqtrrdi.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sseqtrrdi.2 | ⊢ 𝐶 = 𝐵 |
Ref | Expression |
---|---|
sseqtrrdi | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtrrdi.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sseqtrrdi.2 | . . 3 ⊢ 𝐶 = 𝐵 | |
3 | 2 | eqcomi 2181 | . 2 ⊢ 𝐵 = 𝐶 |
4 | 1, 3 | sseqtrdi 3203 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ⊆ wss 3129 |
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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3135 df-ss 3142 |
This theorem is referenced by: iunpw 4476 iotanul 5188 iotass 5190 tfrlem9 6313 tfrlemibfn 6322 tfrlemiubacc 6324 tfrlemi14d 6327 tfr1onlemssrecs 6333 tfr1onlemres 6343 tfrcllemres 6356 exmidfodomrlemr 7194 exmidfodomrlemrALT 7195 uznnssnn 9553 shftfvalg 10798 shftfval 10801 clim2prod 11518 reldvdsrsrg 13073 dvdsrvald 13074 dvdsrex 13079 eltopss 13140 difopn 13241 tgrest 13302 txuni2 13389 tgioo 13679 |
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