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Mirrors > Home > ILE Home > Th. List > pweqd | GIF version |
Description: Equality deduction for power class. (Contributed by NM, 27-Nov-2013.) |
Ref | Expression |
---|---|
pweqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
pweqd | ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | pweq 3577 | . 2 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝒫 𝐴 = 𝒫 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 𝒫 cpw 3574 |
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-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3135 df-ss 3142 df-pw 3576 |
This theorem is referenced by: pmvalg 6653 issubm 12750 basis1 13205 baspartn 13208 cldval 13259 ntrfval 13260 clsfval 13261 neifval 13300 mopnfss 13607 |
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