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Mirrors > Home > MPE Home > Th. List > rabbia2 | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rabbia2.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)) |
Ref | Expression |
---|---|
rabbia2 | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbia2.1 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
3 | 2 | rabbidva2 3423 | . 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
4 | 3 | mptru 1545 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 {crab 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-rab 3115 |
This theorem is referenced by: rabeqi 3429 rabswap 3436 f1ossf1o 6867 finsumvtxdg2ssteplem3 27337 clwlknf1oclwwlkn 27869 clwwlknon2x 27888 numclwwlkovh 28158 ballotlem2 31856 smflim 43410 smflim2 43437 smflimsuplem1 43451 smflimsup 43459 sprvalpwn0 44000 |
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