![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mobid | Structured version Visualization version GIF version |
Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2192, ax-11 2207, ax-13 2389. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.) |
Ref | Expression |
---|---|
mobid.1 | ⊢ Ⅎ𝑥𝜑 |
mobid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
mobid | ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mobid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | mobid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | alrimi 2256 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | mobi 2613 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1654 Ⅎwnf 1882 ∃*wmo 2603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-12 2220 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1879 df-nf 1883 df-mo 2605 |
This theorem is referenced by: eubidOLD 2661 mobidvOLD 2679 moanim 2708 rmobida 3341 rmoeq1f 3349 funcnvmpt 30012 |
Copyright terms: Public domain | W3C validator |