Theorem 19.28 2096

Description: Theorem 19.28 of [Margaris] p. 90. See 19.28v 1909 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.)

Hypothesis

Ref	Expression
19.28.1	|- F/ x ph

Assertion

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

Proof of Theorem 19.28

Step	Hyp	Ref	Expression
1		19.26 1798	. 2 |- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )
2		19.28.1	. . . 4 |- F/ x ph
3	2	19.3 2069	. . 3 |- ( A. x ph <-> ph )
4	3	anbi1i 731	. 2 |- ( ( A. x ph /\ A. x ps ) <-> ( ph /\ A. x ps ) )
5	1, 4	bitri 264	1 |- ( A. x ( ph /\ ps ) <-> ( ph /\ A. x ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  aaan  2170  wl-ax11-lem7  33368