Theorem bnj596 29628

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj596.1	|- ( ph -> A. x ph )
bnj596.2	|- ( ph -> E. x ps )

Assertion

Ref	Expression
bnj596	|- ( ph -> E. x ( ph /\ ps ) )

Proof of Theorem bnj596

Step	Hyp	Ref	Expression
1		bnj596.2	. . 3 |- ( ph -> E. x ps )
2	1	ancli 560	. 2 |- ( ph -> ( ph /\ E. x ps ) )
3		bnj596.1	. . . 4 |- ( ph -> A. x ph )
4	3	nfi 1682	. . 3 |- F/ x ph
5	4	19.42 2071	. 2 |- ( E. x ( ph /\ ps ) <-> ( ph /\ E. x ps ) )
6	2, 5	sylibr 217	1 |- ( ph -> E. x ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   E.wex 1671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676

This theorem is referenced by:  bnj1275  29697  bnj1340  29707  bnj594  29795  bnj1398  29915