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Mirrors > Home > ILE Home > Th. List > 19.26 | GIF version |
Description: Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
Ref | Expression |
---|---|
19.26 | ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | alimi 1344 | . . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → ∀𝑥𝜑) |
3 | simpr 103 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
4 | 3 | alimi 1344 | . . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → ∀𝑥𝜓) |
5 | 2, 4 | jca 290 | . 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → (∀𝑥𝜑 ∧ ∀𝑥𝜓)) |
6 | id 19 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓)) | |
7 | 6 | alanimi 1348 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) |
8 | 5, 7 | impbii 117 | 1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: 19.26-2 1371 19.26-3an 1372 albiim 1376 2albiim 1377 hband 1378 hban 1439 19.27h 1452 19.27 1453 19.28h 1454 19.28 1455 nford 1459 nfand 1460 equsexd 1617 equveli 1642 sbanv 1769 2eu4 1993 bm1.1 2025 r19.26m 2444 unss 3117 ralunb 3124 ssin 3159 intun 3646 intpr 3647 eqrelrel 4441 relop 4486 eqoprab2b 5563 dfer2 6107 |
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