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Mirrors > Home > ILE Home > Th. List > r19.44mv | GIF version |
Description: Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
r19.44mv | ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.9rmv 3357 | . . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | 1 | orbi2d 737 | . 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))) |
3 | r19.43 2520 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓)) | |
4 | 2, 3 | syl6rbbr 197 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∨ wo 662 ∃wex 1424 ∈ wcel 1436 ∃wrex 2356 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-cleq 2078 df-clel 2081 df-rex 2361 |
This theorem is referenced by: frecabcl 6099 |
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