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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 27147. (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 5728 | . 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 5501 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒)) |
6 | 2, 5 | biadanii 821 | 1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 class class class wbr 5110 {copab 5172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 |
This theorem is referenced by: cantnfresb 41688 rp-brsslt 41769 |
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