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Mirrors > Home > MPE Home > Th. List > rabbid | Structured version Visualization version GIF version |
Description: Version of rabbidv 3437 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
rabbid.n | ⊢ Ⅎ𝑥𝜑 |
rabbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rabbid | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbid.n | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | rabbid.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
4 | 1, 3 | rabbida 3455 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 {crab 3429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-9 2109 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-rab 3430 |
This theorem is referenced by: satfv1 34973 bj-rabeqbid 36399 bj-seex 36400 |
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