Metamath Proof Explorer


Theorem 19.38

Description: Theorem 19.38 of Margaris p. 90. The converse holds under nonfreeness conditions, see 19.38a and 19.38b . (Contributed by NM, 12-Mar-1993) Allow a shortening of 19.21t . (Revised by Wolf Lammen, 2-Jan-2018)

Ref Expression
Assertion 19.38 xφxψxφψ

Proof

Step Hyp Ref Expression
1 alnex x¬φ¬xφ
2 pm2.21 ¬φφψ
3 2 alimi x¬φxφψ
4 1 3 sylbir ¬xφxφψ
5 ala1 xψxφψ
6 4 5 ja xφxψxφψ