Theorem r19.32vdc 2503

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

Assertion

Ref	Expression
r19.32vdc	|- (DECID ph -> ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem r19.32vdc

Step	Hyp	Ref	Expression
1		r19.21v 2438	. . 3 |- ( A. x e. A ( -. ph -> ps ) <-> ( -. ph -> A. x e. A ps ) )
2	1	a1i 9	. 2 |- (DECID ph -> ( A. x e. A ( -. ph -> ps ) <-> ( -. ph -> A. x e. A ps ) ) )
3		dfordc 824	. . 3 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )
4	3	ralbidv 2368	. 2 |- (DECID ph -> ( A. x e. A ( ph \/ ps ) <-> A. x e. A ( -. ph -> ps ) ) )
5		dfordc 824	. 2 |- (DECID ph -> ( ( ph \/ A. x e. A ps ) <-> ( -. ph -> A. x e. A ps ) ) )
6	2, 4, 5	3bitr4d 218	1 |- (DECID ph -> ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661  DECID wdc 775   A.wral 2348

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-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-dc 776  df-nf 1390  df-ral 2353

This theorem is referenced by: (None)