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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 3127 | . 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 1331 ⊆ wss 3071 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: sseqtrrd 3136 fssdmd 5286 resasplitss 5302 nnaword2 6410 erssxp 6452 phpm 6759 ioodisj 9776 tgcl 12233 basgen 12249 bastop1 12252 bastop2 12253 clsss2 12298 topssnei 12331 cnntr 12394 txbasval 12436 neitx 12437 cnmpt1res 12465 cnmpt2res 12466 imasnopn 12468 hmeontr 12482 tgioo 12715 reldvg 12817 dvfvalap 12819 dvbss 12823 dvcnp2cntop 12832 dvaddxxbr 12834 dvmulxxbr 12835 dvcj 12842 nninfalllemn 13202 |
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