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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabeqd | Structured version Visualization version GIF version |
Description: Deduction form of rabeq 3485. Note that contrary to rabeq 3485 it has no disjoint variable condition. (Contributed by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
bj-rabeqd.nf | ⊢ Ⅎ𝑥𝜑 |
bj-rabeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
bj-rabeqd | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-rabeqd.nf | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | bj-rabeqd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | eleq2 2903 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
4 | 3 | anbi1d 631 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) |
6 | 1, 5 | bj-rabbida2 34239 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 {crab 3144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-rab 3149 |
This theorem is referenced by: bj-rabeqbid 34241 bj-rabeqbida 34242 bj-inrab2 34248 |
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