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Mirrors > Home > ILE Home > Th. List > raltp | GIF version |
Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
raltp.1 | ⊢ 𝐴 ∈ V |
raltp.2 | ⊢ 𝐵 ∈ V |
raltp.3 | ⊢ 𝐶 ∈ V |
raltp.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
raltp.5 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
raltp.6 | ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃)) |
Ref | Expression |
---|---|
raltp | ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raltp.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | raltp.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | raltp.3 | . 2 ⊢ 𝐶 ∈ V | |
4 | raltp.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | raltp.5 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
6 | raltp.6 | . . 3 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃)) | |
7 | 4, 5, 6 | raltpg 3663 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))) |
8 | 1, 2, 3, 7 | mp3an 1348 | 1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∀wral 2468 Vcvv 2752 {ctp 3612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-sbc 2978 df-un 3148 df-sn 3616 df-pr 3617 df-tp 3618 |
This theorem is referenced by: fztpval 10119 |
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