Theorem 19.32dc 1657

Description: Theorem 19.32 of [Margaris] p. 90, where ph is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)

Hypothesis

Ref	Expression
19.32dc.1	|- F/ x ph

Assertion

Ref	Expression
19.32dc	|- (DECID ph -> ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) )

Proof of Theorem 19.32dc

Step	Hyp	Ref	Expression
1		19.32dc.1	. . . . 5 |- F/ x ph
2	1	nfn 1636	. . . 4 |- F/ x -. ph
3	2	19.21 1562	. . 3 |- ( A. x ( -. ph -> ps ) <-> ( -. ph -> A. x ps ) )
4	3	a1i 9	. 2 |- (DECID ph -> ( A. x ( -. ph -> ps ) <-> ( -. ph -> A. x ps ) ) )
5	1	nfdc 1637	. . 3 |- F/ xDECID ph
6		dfordc 877	. . 3 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )
7	5, 6	albid 1594	. 2 |- (DECID ph -> ( A. x ( ph \/ ps ) <-> A. x ( -. ph -> ps ) ) )
8		dfordc 877	. 2 |- (DECID ph -> ( ( ph \/ A. x ps ) <-> ( -. ph -> A. x ps ) ) )
9	4, 7, 8	3bitr4d 219	1 |- (DECID ph -> ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 697  DECID wdc 819   A.wal 1329   F/wnf 1436

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437

This theorem is referenced by: (None)