Theorem 19.21h

Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " \( x \) is not free in \( \varphi \) ". See also 19.21 and 19.21v. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)

Hypothesis

19.21h.1

\( \vdash \varphi \rightarrow \forall x \, \varphi \)

Conclusion

\( \vdash \forall x \, (\varphi \rightarrow \psi) \leftrightarrow \varphi \rightarrow \forall x \, \psi \)

Proof

19.21h.1

nf5i _

19.21 _