Theorem r19.26m 3067

Description: Version of 19.26 1798 and r19.26 3064 with restricted quantifiers ranging over different classes. (Contributed by NM, 22-Feb-2004.)

Assertion

Ref	Expression
r19.26m	|- ( A. x ( ( x e. A -> ph ) /\ ( x e. B -> ps ) ) <-> ( A. x e. A ph /\ A. x e. B ps ) )

Proof of Theorem r19.26m

Step	Hyp	Ref	Expression
1		19.26 1798	. 2 |- ( A. x ( ( x e. A -> ph ) /\ ( x e. B -> ps ) ) <-> ( A. x ( x e. A -> ph ) /\ A. x ( x e. B -> ps ) ) )
2		df-ral 2917	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3		df-ral 2917	. . 3 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
4	2, 3	anbi12i 733	. 2 |- ( ( A. x e. A ph /\ A. x e. B ps ) <-> ( A. x ( x e. A -> ph ) /\ A. x ( x e. B -> ps ) ) )
5	1, 4	bitr4i 267	1 |- ( A. x ( ( x e. A -> ph ) /\ ( x e. B -> ps ) ) <-> ( A. x e. A ph /\ A. x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by: (None)