Theorem r19.23t 3021

Description: Closed theorem form of r19.23 3022. (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 2079	. 2 |- ( F/ x ps -> ( A. x ( ( x e. A /\ ph ) -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) ) )
2		df-ral 2917	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
3		impexp 462	. . . 4 |- ( ( ( x e. A /\ ph ) -> ps ) <-> ( x e. A -> ( ph -> ps ) ) )
4	3	albii 1747	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
5	2, 4	bitr4i 267	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ps ) )
6		df-rex 2918	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
7	6	imbi1i 339	. 2 |- ( ( E. x e. A ph -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) )
8	1, 5, 7	3bitr4g 303	1 |- ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  r19.23  3022  rexlimd2  3025  riotasv3d  34246