Theorem 19.38a 1767

Description: Under a non-freeness hypothesis, the implication 19.38 1766 can be strengthened to an equivalence. See also 19.38b 1768. (Contributed by BJ, 3-Nov-2021.)

Assertion

Ref	Expression
19.38a	|- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )

Proof of Theorem 19.38a

Step	Hyp	Ref	Expression
1		19.38 1766	. 2 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
2		df-nf 1710	. . 3 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
3		alim 1738	. . . 4 |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
4		imim1 83	. . . 4 |- ( ( E. x ph -> A. x ph ) -> ( ( A. x ph -> A. x ps ) -> ( E. x ph -> A. x ps ) ) )
5	3, 4	syl5 34	. . 3 |- ( ( E. x ph -> A. x ph ) -> ( A. x ( ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
6	2, 5	sylbi 207	. 2 |- ( F/ x ph -> ( A. x ( ph -> ps ) -> ( E. x ph -> A. x ps ) ) )
7	1, 6	impbid2 216	1 |- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> A. x ( ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  19.21t  2073