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Mirrors > Home > MPE Home > Th. List > pweqALT | Structured version Visualization version GIF version |
Description: Alternate proof of pweq 4611 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 4003 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵)) | |
2 | 1 | abbidv 2795 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ 𝑥 ⊆ 𝐴} = {𝑥 ∣ 𝑥 ⊆ 𝐵}) |
3 | df-pw 4599 | . 2 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
4 | df-pw 4599 | . 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵} | |
5 | 2, 3, 4 | 3eqtr4g 2791 | 1 ⊢ (𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {cab 2703 ⊆ wss 3943 𝒫 cpw 4597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-ss 3960 df-pw 4599 |
This theorem is referenced by: (None) |
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