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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 3354 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) | |
2 | 1 | orbi1d 738 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ((𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))) |
3 | r19.43 2518 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | |
4 | 2, 3 | syl6rbbr 197 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∨ wo 662 ∃wex 1422 ∈ wcel 1434 ∃wrex 2354 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-cleq 2076 df-clel 2079 df-rex 2359 |
This theorem is referenced by: ltexprlemloc 7069 |
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