Theorem 19.32dc 1609

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 1588	. . . 4 |- F/ x -. ph
3	2	19.21 1515	. . 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 1589	. . 3 |- F/ xDECID ph
6		dfordc 824	. . 3 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )
7	5, 6	albid 1546	. 2 |- (DECID ph -> ( A. x ( ph \/ ps ) <-> A. x ( -. ph -> ps ) ) )
8		dfordc 824	. 2 |- (DECID ph -> ( ( ph \/ A. x ps ) <-> ( -. ph -> A. x ps ) ) )
9	4, 7, 8	3bitr4d 218	1 |- (DECID ph -> ( A. x ( ph \/ ps ) <-> ( ph \/ A. x ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661  DECID wdc 775   A.wal 1282   F/wnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-fal 1290  df-nf 1390

This theorem is referenced by: (None)