Theorem 19.45 2107

Description: Theorem 19.45 of [Margaris] p. 90. See 19.45v 1913 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)

Hypothesis

Ref	Expression
19.45.1	|- F/ x ph

Assertion

Ref	Expression
19.45	|- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) )

Proof of Theorem 19.45

Step	Hyp	Ref	Expression
1		19.43 1810	. 2 |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )
2		19.45.1	. . . 4 |- F/ x ph
3	2	19.9 2072	. . 3 |- ( E. x ph <-> ph )
4	3	orbi1i 542	. 2 |- ( ( E. x ph \/ E. x ps ) <-> ( ph \/ E. x ps ) )
5	1, 4	bitri 264	1 |- ( E. x ( ph \/ ps ) <-> ( ph \/ E. x ps ) )

Syntax hints:   <-> wb 196   \/ wo 383   E.wex 1704   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-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  eeor  2171