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| Mirrors > Home > MPE Home > Th. List > pweqALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of pweq 4614 directly from the definition. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| pweqALT | ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sseq2 4010 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵)) | |
| 2 | 1 | abbidv 2808 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵}) |
| 3 | df-pw 4602 | . 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
| 4 | df-pw 4602 | . 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵} | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 {cab 2714 ⊆ wss 3951 𝒫 cpw 4600 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-ss 3968 df-pw 4602 |
| This theorem is referenced by: (None) |
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