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Mirrors > Home > ILE Home > Th. List > sseq1d | GIF version |
Description: An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
sseq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sseq1d | ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | sseq1 3180 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ⊆ wss 3131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3137 df-ss 3144 |
This theorem is referenced by: sseq12d 3188 eqsstrd 3193 snssgOLD 3730 ssiun2s 3932 treq 4109 onsucsssucexmid 4528 funimass1 5295 feq1 5350 sbcfg 5366 fvmptssdm 5603 fvimacnvi 5633 nnsucsssuc 6496 ereq1 6545 elpm2r 6669 fipwssg 6981 nnnninf 7127 ctssexmid 7151 rspssp 13612 iscnp 13887 iscnp4 13906 cnntr 13913 cnconst2 13921 cnptopresti 13926 cnptoprest 13927 txbas 13946 txcnp 13959 txdis 13965 txdis1cn 13966 blssps 14115 blss 14116 ssblex 14119 blin2 14120 metss2 14186 metrest 14194 metcnp3 14199 cnopnap 14282 limccl 14316 ellimc3apf 14317 |
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