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Mirrors > Home > ILE Home > Th. List > r19.45mv | GIF version |
Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
r19.45mv | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.9rmv 3377 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) | |
2 | 1 | orbi1d 741 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ((𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))) |
3 | r19.43 2526 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | |
4 | 2, 3 | syl6rbbr 198 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 665 ∃wex 1427 ∈ wcel 1439 ∃wrex 2361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-cleq 2082 df-clel 2085 df-rex 2366 |
This theorem is referenced by: ltexprlemloc 7227 |
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