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Mirrors > Home > MPE Home > Th. List > eloprabg | Structured version Visualization version GIF version |
Description: The law of concretion for operation class abstraction. Compare elopab 4983. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
eloprabg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
eloprabg.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
eloprabg.3 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) |
Ref | Expression |
---|---|
eloprabg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloprabg.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | eloprabg.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | eloprabg.3 | . . 3 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
4 | 1, 2, 3 | syl3an9b 1397 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜃)) |
5 | 4 | eloprabga 6747 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 〈cop 4183 {coprab 6651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-oprab 6654 |
This theorem is referenced by: ov 6780 ovg 6799 brbtwn 25779 isnvlem 27465 isphg 27672 fvtransport 32139 brcolinear2 32165 colineardim1 32168 fvray 32248 fvline 32251 |
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