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Mirrors > Home > ILE Home > Th. List > 19.41 | GIF version |
Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
Ref | Expression |
---|---|
19.41.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
19.41 | ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.40 1611 | . . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓)) | |
2 | 19.41.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.9 1624 | . . . 4 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 3 | anbi2i 453 | . . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
5 | 1, 4 | sylib 121 | . 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ 𝜓)) |
6 | pm3.21 262 | . . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
7 | 2, 6 | eximd 1592 | . . 3 ⊢ (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
8 | 7 | impcom 124 | . 2 ⊢ ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
9 | 5, 8 | impbii 125 | 1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 Ⅎwnf 1440 ∃wex 1472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-nf 1441 |
This theorem is referenced by: 19.42 1668 eean 1911 r19.41 2612 eliunxp 4725 dfopab2 6137 dfoprab3s 6138 xpcomco 6771 |
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