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| Mirrors > Home > MPE Home > Th. List > rabeqbidva | Structured version Visualization version GIF version | ||
| Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove DV conditions. (Revised by GG, 1-Sep-2025.) | 
| Ref | Expression | 
|---|---|
| rabeqbidva.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| rabeqbidva.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rabeqbidva | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rabeqbidva.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | rabbidva 3442 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) | 
| 3 | rabeqbidva.1 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 4 | 3 | eleq2d 2826 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | 
| 5 | 4 | anbi1d 631 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜒) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) | 
| 6 | 5 | rabbidva2 3437 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | 
| 7 | 2, 6 | eqtrd 2776 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 | 
| This theorem is referenced by: rabeqbidv 3454 natpropd 18025 gsumpropd2lem 18693 elntg 29000 rmfsupp2 33243 poimirlem28 37656 scotteqd 44261 uspgrlimlem1 47960 domnmsuppn0 48290 eenglngeehlnm 48665 | 
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