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Mirrors > Home > ILE Home > Th. List > r19.44av | GIF version |
Description: One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when 𝐴 is empty. (Contributed by NM, 2-Apr-2004.) |
Ref | Expression |
---|---|
r19.44av | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.43 2615 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | idd 21 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜓 → 𝜓)) | |
3 | 2 | rexlimiv 2568 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 → 𝜓) |
4 | 3 | orim2i 751 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)) |
5 | 1, 4 | sylbi 120 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 ∈ wcel 2128 ∃wrex 2436 |
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-io 699 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-17 1506 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-ral 2440 df-rex 2441 |
This theorem is referenced by: (None) |
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