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Mirrors > Home > MPE Home > Th. List > rabbida | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3437 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2135, ax-11 2152. (Revised by Wolf Lammen, 14-Mar-2025.) |
Ref | Expression |
---|---|
rabbida.n | ⊢ Ⅎ𝑥𝜑 |
rabbida.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rabbida | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbida.n | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | rabbida.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | pm5.32da 577 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | 1, 3 | rabbida4 3455 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 Ⅎwnf 1783 ∈ 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-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-rab 3431 |
This theorem is referenced by: rabbid 3457 rabeqbida 3459 smfpimltmpt 45760 smfpimltxrmptf 45772 smfpimgtmpt 45795 smfpimgtxrmptf 45798 smfrec 45803 smfsupmpt 45829 smfinflem 45831 smfinfmpt 45833 |
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