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Mirrors > Home > ILE Home > Th. List > eqsstrrid | GIF version |
Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
eqsstrrid.1 | ⊢ 𝐵 = 𝐴 |
eqsstrrid.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
eqsstrrid | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstrrid.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | eqcomi 2174 | . 2 ⊢ 𝐴 = 𝐵 |
3 | eqsstrrid.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | eqsstrid 3193 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ⊆ wss 3121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-in 3127 df-ss 3134 |
This theorem is referenced by: abnexg 4431 relcnvtr 5130 resasplitss 5377 fimacnvdisj 5382 fimacnv 5625 f1ompt 5647 tfr1onlemres 6328 tfrcllemres 6341 |
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