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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 27284. (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 5767 | . 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 5539 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒)) |
6 | 2, 5 | biadanii 820 | 1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 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 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 |
This theorem is referenced by: cantnfresb 42064 rp-brsslt 42164 |
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