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Mirrors > Home > MPE Home > Th. List > euanv | Structured version Visualization version GIF version |
Description: Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.) |
Ref | Expression |
---|---|
euanv | ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euex 2637 | . . . 4 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | |
2 | simpl 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
3 | 2 | exlimiv 1931 | . . . 4 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → 𝜑) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → 𝜑) |
5 | ibar 532 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ∧ 𝜓))) | |
6 | 5 | eubidv 2647 | . . . 4 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∧ 𝜓))) |
7 | 6 | biimprcd 253 | . . 3 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 → ∃!𝑥𝜓)) |
8 | 4, 7 | jcai 520 | . 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → (𝜑 ∧ ∃!𝑥𝜓)) |
9 | 6 | biimpa 480 | . 2 ⊢ ((𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∧ 𝜓)) |
10 | 8, 9 | impbii 212 | 1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1781 ∃!weu 2628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-mo 2598 df-eu 2629 |
This theorem is referenced by: eueq2 3649 2reu5lem1 3694 fsn 6874 dfac5lem5 9538 |
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