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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 27845. (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 5780 | . 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 5549 | . 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 1537 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 {copab 5210 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 |
This theorem is referenced by: cantnfresb 43314 rp-brsslt 43413 |
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