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| Mirrors > Home > MPE Home > Th. List > opelopab | Structured version Visualization version GIF version | ||
| Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.) |
| Ref | Expression |
|---|---|
| opelopab.1 | ⊢ 𝐴 ∈ V |
| opelopab.2 | ⊢ 𝐵 ∈ V |
| opelopab.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| opelopab.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| opelopab | ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opelopab.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | opelopab.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | opelopab.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 4 | opelopab.4 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 5 | 3, 4 | opelopabg 5543 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) |
| 6 | 1, 2, 5 | mp2an 692 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 {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-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-opab 5206 |
| This theorem is referenced by: opabid2 5838 dfres2 6059 f1oiso 7371 elopabi 8087 xporderlem 8152 cnlnssadj 32099 areacirclem5 37719 dicopelval 41179 dih1dimatlem 41331 pellexlem3 42842 fsovrfovd 44022 |
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