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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 3432 | . 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
4 | 3 | mptru 1546 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1539 ⊤wtru 1540 ∈ wcel 2104 {crab 3430 |
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 1911 ax-6 1969 ax-7 2009 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-rab 3431 |
This theorem is referenced by: rabbiia 3434 rabswap 3439 rabeqi 3443 rabrabi 3448 f1ossf1o 7127 finsumvtxdg2ssteplem3 29071 clwlknf1oclwwlkn 29604 clwwlknon2x 29623 numclwwlkovh 29893 ballotlem2 33785 smflim 45791 smflim2 45820 smflimsuplem1 45834 smflimsup 45842 sprvalpwn0 46449 |
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