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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-rabeqd | Structured version Visualization version GIF version |
Description: Deduction form of rabeq 3486. Note that contrary to rabeq 3486 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 2904 | . . . 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 34241 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 {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-8 2115 ax-9 2123 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-rab 3150 |
This theorem is referenced by: bj-rabeqbid 34243 bj-rabeqbida 34244 bj-inrab2 34250 |
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