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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bropabg | Structured version Visualization version GIF version | ||
| Description: Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27830. (Contributed by RP, 26-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| bropabg.xA | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| bropabg.yB | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | 
| bropabg.R | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | 
| Ref | Expression | 
|---|---|
| bropabg | ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bropabg.R | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 2 | 1 | bropaex12 5777 | . 2 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| 3 | bropabg.xA | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 4 | bropabg.yB | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 5 | 3, 4, 1 | brabg 5544 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒)) | 
| 6 | 2, 5 | biadanii 822 | 1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 | 
| This theorem is referenced by: cantnfresb 43337 rp-brsslt 43436 | 
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