![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rexlimdvv | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.) |
Ref | Expression |
---|---|
rexlimdvv.1 | ⊢ (φ → ((x ∈ A ∧ y ∈ B) → (ψ → χ))) |
Ref | Expression |
---|---|
rexlimdvv | ⊢ (φ → (∃x ∈ A ∃y ∈ B ψ → χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimdvv.1 | . . . 4 ⊢ (φ → ((x ∈ A ∧ y ∈ B) → (ψ → χ))) | |
2 | 1 | expdimp 246 | . . 3 ⊢ ((φ ∧ x ∈ A) → (y ∈ B → (ψ → χ))) |
3 | 2 | rexlimdv 2426 | . 2 ⊢ ((φ ∧ x ∈ A) → (∃y ∈ B ψ → χ)) |
4 | 3 | rexlimdva 2427 | 1 ⊢ (φ → (∃x ∈ A ∃y ∈ B ψ → χ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 ∃wrex 2301 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-ial 1424 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-ral 2305 df-rex 2306 |
This theorem is referenced by: rexlimdvva 2434 f1oiso2 5409 xpdom2 6241 genpcdl 6502 genpcuu 6503 distrlem1prl 6558 distrlem1pru 6559 distrlem5prl 6562 distrlem5pru 6563 recexprlemss1l 6607 recexprlemss1u 6608 qaddcl 8346 qmulcl 8348 |
Copyright terms: Public domain | W3C validator |