Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > euor2 | Structured version Visualization version GIF version |
Description: Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) |
Ref | Expression |
---|---|
euor2 | ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2145 | . . 3 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
2 | 1 | nfn 1848 | . 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑 |
3 | 19.8a 2170 | . . . 4 ⊢ (𝜑 → ∃𝑥𝜑) | |
4 | 3 | con3i 157 | . . 3 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑) |
5 | biorf 930 | . . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
6 | 5 | bicomd 224 | . . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
7 | 4, 6 | syl 17 | . 2 ⊢ (¬ ∃𝑥𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
8 | 2, 7 | eubid 2666 | 1 ⊢ (¬ ∃𝑥𝜑 → (∃!𝑥(𝜑 ∨ 𝜓) ↔ ∃!𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∨ wo 841 ∃wex 1771 ∃!weu 2646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-mo 2615 df-eu 2647 |
This theorem is referenced by: reuun2 4283 |
Copyright terms: Public domain | W3C validator |