Theorem r19.32vdc 2626

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 2554	. . 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 892	. . 3 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )
4	3	ralbidv 2477	. 2 |- (DECID ph -> ( A. x e. A ( ph \/ ps ) <-> A. x e. A ( -. ph -> ps ) ) )
5		dfordc 892	. 2 |- (DECID ph -> ( ( ph \/ A. x e. A ps ) <-> ( -. ph -> A. x e. A ps ) ) )
6	2, 4, 5	3bitr4d 220	1 |- (DECID ph -> ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 708  DECID wdc 834   A.wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-dc 835  df-nf 1461  df-ral 2460

This theorem is referenced by: (None)