Theorem r19.23t 2423

Description: Closed theorem form of r19.23 2424. (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 1567	. 2 |- ( F/ x ps -> ( A. x ( ( x e. A /\ ph ) -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) ) )
2		df-ral 2311	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
3		impexp 250	. . . 4 |- ( ( ( x e. A /\ ph ) -> ps ) <-> ( x e. A -> ( ph -> ps ) ) )
4	3	albii 1359	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
5	2, 4	bitr4i 176	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ( x e. A /\ ph ) -> ps ) )
6		df-rex 2312	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
7	6	imbi1i 227	. 2 |- ( ( E. x e. A ph -> ps ) <-> ( E. x ( x e. A /\ ph ) -> ps ) )
8	1, 5, 7	3bitr4g 212	1 |- ( F/ x ps -> ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381   e. wcel 1393   A.wral 2306   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  r19.23  2424  rexlimd2  2431