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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 3425 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
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 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rabbida.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | ex 416 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒))) |
4 | 1, 3 | ralrimi 3180 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒)) |
5 | rabbi 3336 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | |
6 | 4, 5 | sylib 221 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 {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-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-ral 3111 df-rab 3115 |
This theorem is referenced by: rabbid 3422 bj-rabeqbida 34364 pimgtmnf 43357 smfpimltmpt 43380 smfpimltxrmpt 43392 smfpimgtmpt 43414 smfpimgtxrmpt 43417 smfrec 43421 smfsupmpt 43446 smfinflem 43448 smfinfmpt 43450 |
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