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Mirrors > Home > ILE Home > Th. List > r19.45av | GIF version |
Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.) |
Ref | Expression |
---|---|
r19.45av | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.43 2628 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | idd 21 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜑)) | |
3 | 2 | rexlimiv 2581 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜑) |
4 | 3 | orim1i 755 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
5 | 1, 4 | sylbi 120 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 703 ∈ wcel 2141 ∃wrex 2449 |
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 704 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-ial 1527 ax-i5r 1528 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-ral 2453 df-rex 2454 |
This theorem is referenced by: (None) |
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