![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 19.42vv | Structured version Visualization version GIF version |
Description: Version of 19.42 2236 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.) |
Ref | Expression |
---|---|
19.42vv | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exdistr 1955 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) | |
2 | 19.42v 1954 | . 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓)) | |
3 | 1, 2 | bitri 278 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1781 |
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 |
This theorem is referenced by: exdistr2 1959 19.42vvvOLD 1961 3exdistr 1962 ceqsex3v 3493 ceqsex4v 3494 ceqsex8v 3496 elvvv 5591 xpdifid 5992 dfoprab2 7191 resoprab 7249 elrnmpores 7267 ov3 7291 ov6g 7292 oprabex3 7660 xpassen 8594 axaddf 10556 axmulf 10557 qqhval2 31333 bnj996 32338 inxpxrn 35803 dvhopellsm 38413 |
Copyright terms: Public domain | W3C validator |