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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 5528. (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 1435 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜃)) |
5 | 4 | eloprabga 7516 | 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 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⟨cop 4635 {coprab 7410 |
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-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-oprab 7413 |
This theorem is referenced by: ov 7552 ovg 7572 brbtwn 28157 isnvlem 29863 isphg 30070 fvtransport 35004 brcolinear2 35030 colineardim1 35033 fvray 35113 fvline 35116 |
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