Theorem bnj1345 30895

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

Hypotheses

Ref	Expression
bnj1345.1	|- ( ph -> E. x ( ps /\ ch ) )
bnj1345.2	|- ( th <-> ( ph /\ ps /\ ch ) )
bnj1345.3	|- ( ph -> A. x ph )

Assertion

Ref	Expression
bnj1345	|- ( ph -> E. x th )

Proof of Theorem bnj1345

Step	Hyp	Ref	Expression
1		bnj1345.1	. . 3 |- ( ph -> E. x ( ps /\ ch ) )
2		bnj1345.3	. . 3 |- ( ph -> A. x ph )
3	1, 2	bnj1275 30884	. 2 |- ( ph -> E. x ( ph /\ ps /\ ch ) )
4		bnj1345.2	. 2 |- ( th <-> ( ph /\ ps /\ ch ) )
5	3, 4	bnj1198 30866	1 |- ( ph -> E. x th )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   E.wex 1704

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-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705  df-nf 1710

This theorem is referenced by:  bnj1379  30901  bnj1521  30921