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Mirrors > Home > ILE Home > Th. List > syl6sseq | GIF version |
Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
syl6sseq.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
syl6sseq.2 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
syl6sseq | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6sseq.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | syl6sseq.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 2 | sseq2i 3051 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶) |
4 | 1, 3 | sylib 120 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ⊆ wss 2999 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-11 1442 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-in 3005 df-ss 3012 |
This theorem is referenced by: syl6sseqr 3073 onintonm 4332 relrelss 4952 iotanul 4990 foimacnv 5265 cauappcvgprlemladdru 7205 zisum 10761 fsum3cvg3 10776 |
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