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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 3481 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 415 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒))) |
4 | 1, 3 | ralrimi 3219 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒)) |
5 | rabbi 3386 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | |
6 | 4, 5 | sylib 220 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 ∀wral 3141 {crab 3145 |
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 1910 ax-6 1969 ax-7 2014 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-ral 3146 df-rab 3150 |
This theorem is referenced by: rabbid 3478 bj-rabeqbida 34244 pimgtmnf 43007 smfpimltmpt 43030 smfpimltxrmpt 43042 smfpimgtmpt 43064 smfpimgtxrmpt 43067 smfrec 43071 smfsupmpt 43096 smfinflem 43098 smfinfmpt 43100 |
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