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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 2224. See r19.41v 3179 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 3061 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
2 | anass 467 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) | |
3 | 2 | exbii 1843 | . 2 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓))) |
4 | r19.41.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | 19.41 2224 | . . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓)) |
6 | df-rex 3061 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
7 | 6 | bicomi 223 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑) |
8 | 5, 7 | bianbi 625 | . 2 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) |
9 | 1, 3, 8 | 3bitr2i 298 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∃wex 1774 Ⅎwnf 1778 ∈ wcel 2099 ∃wrex 3060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-nf 1779 df-rex 3061 |
This theorem is referenced by: reuxfrdf 32419 iunin1f 32478 2reu8i 46726 |
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