Theorem 2albiim
Split a biconditional and distribute two quantifiers. (Contributed by NM,
3-Feb-2005.)
Conclusion
\( \vdash \forall x \, \forall y \, (\varphi \leftrightarrow \psi) \leftrightarrow \forall x \, \forall y \, (\varphi \rightarrow \psi) \wedge \forall x \, \forall y \, (\psi \rightarrow \varphi) \)
Proof
albiim
l1
albii _
\( \vdash \forall x \, \forall y \, (\varphi \leftrightarrow \psi) \leftrightarrow \forall x \, (\forall y \, (\varphi \rightarrow \psi) \wedge \forall y \, (\psi \rightarrow \varphi)) \)
19.26
\( \vdash \forall x \, (\forall y \, (\varphi \rightarrow \psi) \wedge \forall y \, (\psi \rightarrow \varphi)) \leftrightarrow \forall x \, \forall y \, (\varphi \rightarrow \psi) \wedge \forall x \, \forall y \, (\psi \rightarrow \varphi) \)
bitri l1 _