Theorem 19.27h 1548

Description: Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.27h.1	|- ( ps -> A. x ps )

Assertion

Ref	Expression
19.27h	|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) )

Proof of Theorem 19.27h

Step	Hyp	Ref	Expression
1		19.26 1469	. 2 |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )
2		19.27h.1	. . . 4 |- ( ps -> A. x ps )
3	2	19.3h 1541	. . 3 |- ( A. x ps <-> ps )
4	3	anbi2i 453	. 2 |- ( ( A. x ph /\ A. x ps ) <-> ( A. x ph /\ ps ) )
5	1, 4	bitri 183	1 |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341

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  ax-4 1498

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  aaanh  1574  19.27v  1887