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Mirrors > Home > NFE 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 3777 | . 2 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (∀x ∈ {A, B, C}φ ↔ (ψ ∧ χ ∧ θ))) |
8 | 1, 2, 3, 7 | mp3an 1277 | 1 ⊢ (∀x ∈ {A, B, C}φ ↔ (ψ ∧ χ ∧ θ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ w3a 934 = wceq 1642 ∈ wcel 1710 ∀wral 2614 Vcvv 2859 {ctp 3739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 df-tp 3743 |
This theorem is referenced by: (None) |
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