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Mirrors > Home > ILE Home > Th. List > sseqtrdi | GIF version |
Description: A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
sseqtrdi.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sseqtrdi.2 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
sseqtrdi | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtrdi.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sseqtrdi.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 2 | sseq2i 3164 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶) |
4 | 1, 3 | sylib 121 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ⊆ wss 3111 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-in 3117 df-ss 3124 |
This theorem is referenced by: sseqtrrdi 3186 onintonm 4488 relrelss 5124 iotanul 5162 foimacnv 5444 cauappcvgprlemladdru 7588 zsumdc 11311 fsum3cvg3 11323 zproddc 11506 nninfdcex 11871 zsupssdc 11872 distop 12626 cnptoprest 12780 pwle2 13712 pw1nct 13717 |
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