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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 5379. (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 1431 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜃)) |
5 | 4 | eloprabga 7240 | 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 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 〈cop 4531 {coprab 7136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-oprab 7139 |
This theorem is referenced by: ov 7273 ovg 7293 brbtwn 26693 isnvlem 28393 isphg 28600 fvtransport 33606 brcolinear2 33632 colineardim1 33635 fvray 33715 fvline 33718 |
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