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Mirrors > Home > ILE Home > Th. List > sseqtrd | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) |
Ref | Expression |
---|---|
sseqtrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sseqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sseqtrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sseqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 2 | sseq2d 3055 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐶)) |
4 | 1, 3 | mpbid 146 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ⊆ wss 3000 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-11 1443 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-in 3006 df-ss 3013 |
This theorem is referenced by: sseqtr4d 3064 fssdmd 5187 resasplitss 5203 nnaword2 6287 erssxp 6329 phpm 6635 ioodisj 9464 tgcl 11818 basgen 11834 bastop1 11837 bastop2 11838 clsss2 11883 nninfalllemn 12164 |
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