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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-brab2a1 | Structured version Visualization version GIF version |
Description: "Unbounded" version of brab2a 5637. (Contributed by BJ, 25-Dec-2023.) |
Ref | Expression |
---|---|
bj-brab2a1.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
bj-brab2a1.2 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
bj-brab2a1 | ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-brab2a1.1 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
2 | bj-brab2a1.2 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | vex 3494 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3494 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | pm3.2i 473 | . . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
6 | 5 | biantrur 533 | . . . 4 ⊢ (𝜑 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑)) |
7 | 6 | opabbii 5126 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑)} |
8 | 2, 7 | eqtri 2843 | . 2 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝜑)} |
9 | 1, 8 | brab2a 5637 | 1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 class class class wbr 5059 {copab 5121 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 |
This theorem is referenced by: bj-ideqg1 34480 |
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