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| Mirrors > Home > MPE Home > Th. List > r19.41 | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of 19.41 2234. See r19.41v 3188 for a version with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 1-Nov-2010.) | 
| Ref | Expression | 
|---|---|
| r19.41.1 | ⊢ Ⅎ𝑥𝜓 | 
| Ref | Expression | 
|---|---|
| r19.41 | ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rex 3070 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
| 2 | anass 468 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
| 3 | 2 | exbii 1847 | . 2 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | 
| 4 | r19.41.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 5 | 4 | 19.41 2234 | . . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) | 
| 6 | df-rex 3070 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 7 | 6 | bicomi 224 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑) | 
| 8 | 5, 7 | bianbi 627 | . 2 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) | 
| 9 | 1, 3, 8 | 3bitr2i 299 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1778 Ⅎwnf 1782 ∈ wcel 2107 ∃wrex 3069 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-rex 3070 | 
| This theorem is referenced by: reuxfrdf 32511 iunin1f 32571 2reu8i 47130 | 
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