Theorem r19.23t 2729

Description: Closed theorem form of r19.23 2730. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Assertion

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

Proof of Theorem r19.23t

Step	Hyp	Ref	Expression
1		19.23t 1800	. 2 |- ( F/xps -> (A.x((x e. A /\ ph) -> ps) <-> (E.x(x e. A /\ ph) -> ps)))
2		df-ral 2620	. . 3 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
3		impexp 433	. . . 4 |- (((x e. A /\ ph) -> ps) <-> (x e. A -> (ph -> ps)))
4	3	albii 1566	. . 3 |- (A.x((x e. A /\ ph) -> ps) <-> A.x(x e. A -> (ph -> ps)))
5	2, 4	bitr4i 243	. 2 |- (A.x e. A (ph -> ps) <-> A.x((x e. A /\ ph) -> ps))
6		df-rex 2621	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
7	6	imbi1i 315	. 2 |- ((E.x e. A ph -> ps) <-> (E.x(x e. A /\ ph) -> ps))
8	1, 5, 7	3bitr4g 279	1 |- ( F/xps -> (A.x e. A (ph -> ps) <-> (E.x e. A ph -> ps)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544   e. wcel 1710  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  r19.23  2730  rexlimd2  2737