Theorem r19.23t 2542

Description: Closed theorem form of r19.23 2543. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Assertion

Ref	Expression
r19.23t	|- ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )

Proof of Theorem r19.23t

Step	Hyp	Ref	Expression
1		19.23t 1656	. 2 |- ( F/ x ps -> ( A. x ( ( x e. A /\ ph ) -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) ) )
2		df-ral 2422	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
3		impexp 261	. . . 4 |- ( ( ( x e. A /\ ph ) -> ps ) <-> ( x e. A -> ( ph -> ps ) ) )
4	3	albii 1447	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
5	2, 4	bitr4i 186	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ps ) )
6		df-rex 2423	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
7	6	imbi1i 237	. 2 |- ( ( E. x e. A ph -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) )
8	1, 5, 7	3bitr4g 222	1 |- ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   F/wnf 1437   E.wex 1469   e. wcel 1481   A.wral 2417   E.wrex 2418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2422  df-rex 2423

This theorem is referenced by:  r19.23  2543  rexlimd2  2550