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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabeqbida | Structured version Visualization version GIF version |
Description: Version of rabeqbidva 3486 with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
bj-rabeqbida.nf | ⊢ Ⅎ𝑥𝜑 |
bj-rabeqbida.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
bj-rabeqbida.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
bj-rabeqbida | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-rabeqbida.nf | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | bj-rabeqbida.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | rabbida 3474 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
4 | bj-rabeqbida.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | 1, 4 | bj-rabeqd 34238 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
6 | 3, 5 | eqtrd 2856 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 {crab 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-rab 3147 |
This theorem is referenced by: (None) |
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