Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - start with the Axiom of Extensionality
Restricted quantification
nrexralim
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Metamath Proof Explorer
Ascii
Unicode
Theorem
nrexralim
Description:
Negation of a complex predicate calculus formula.
(Contributed by
FL
, 31-Jul-2009)
Ref
Expression
Assertion
nrexralim
⊢
¬
∃
x
∈
A
∀
y
∈
B
φ
→
ψ
↔
∀
x
∈
A
∃
y
∈
B
φ
∧
¬
ψ
Proof
Step
Hyp
Ref
Expression
1
rexanali
⊢
∃
y
∈
B
φ
∧
¬
ψ
↔
¬
∀
y
∈
B
φ
→
ψ
2
1
ralbii
⊢
∀
x
∈
A
∃
y
∈
B
φ
∧
¬
ψ
↔
∀
x
∈
A
¬
∀
y
∈
B
φ
→
ψ
3
ralnex
⊢
∀
x
∈
A
¬
∀
y
∈
B
φ
→
ψ
↔
¬
∃
x
∈
A
∀
y
∈
B
φ
→
ψ
4
2
3
bitr2i
⊢
¬
∃
x
∈
A
∀
y
∈
B
φ
→
ψ
↔
∀
x
∈
A
∃
y
∈
B
φ
∧
¬
ψ