Theorem r19.23t 2604

Description: Closed theorem form of r19.23 2605. (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 1691	. 2 |- ( F/ x ps -> ( A. x ( ( x e. A /\ ph ) -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) ) )
2		df-ral 2480	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
3		impexp 263	. . . 4 |- ( ( ( x e. A /\ ph ) -> ps ) <-> ( x e. A -> ( ph -> ps ) ) )
4	3	albii 1484	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
5	2, 4	bitr4i 187	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ps ) )
6		df-rex 2481	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
7	6	imbi1i 238	. 2 |- ( ( E. x e. A ph -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) )
8	1, 5, 7	3bitr4g 223	1 |- ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   F/wnf 1474   E.wex 1506   e. wcel 2167   A.wral 2475   E.wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480  df-rex 2481

This theorem is referenced by:  r19.23  2605  rexlimd2  2612