Theorem r19.32vdc 2615

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 2543	. . 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 882	. . 3 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )
4	3	ralbidv 2466	. 2 |- (DECID ph -> ( A. x e. A ( ph \/ ps ) <-> A. x e. A ( -. ph -> ps ) ) )
5		dfordc 882	. 2 |- (DECID ph -> ( ( ph \/ A. x e. A ps ) <-> ( -. ph -> A. x e. A ps ) ) )
6	2, 4, 5	3bitr4d 219	1 |- (DECID ph -> ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 698  DECID wdc 824   A.wral 2444

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-dc 825  df-nf 1449  df-ral 2449

This theorem is referenced by: (None)