Theorem r19.26m 2597

Description: Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (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 1469	. 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 2449	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3		df-ral 2449	. . 3 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
4	2, 3	anbi12i 456	. 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 186	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   /\ wa 103   <-> wb 104   A.wal 1341   e. wcel 2136   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-5 1435  ax-gen 1437

This theorem depends on definitions:  df-bi 116  df-ral 2449

This theorem is referenced by: (None)