Theorem bnj1023 30851

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

Hypotheses

Ref	Expression
bnj1023.1	|- E. x ( ph -> ps )
bnj1023.2	|- ( ps -> ch )

Assertion

Ref	Expression
bnj1023	|- E. x ( ph -> ch )

Proof of Theorem bnj1023

Step	Hyp	Ref	Expression
1		bnj1023.2	. . . . 5 |- ( ps -> ch )
2	1	a1i 11	. . . 4 |- ( ( ph -> ps ) -> ( ps -> ch ) )
3	2	ax-gen 1722	. . 3 |- A. x ( ( ph -> ps ) -> ( ps -> ch ) )
4		bnj1023.1	. . 3 |- E. x ( ph -> ps )
5		exintr 1819	. . 3 |- ( A. x ( ( ph -> ps ) -> ( ps -> ch ) ) -> ( E. x ( ph -> ps ) -> E. x ( ( ph -> ps ) /\ ( ps -> ch ) ) ) )
6	3, 4, 5	mp2 9	. 2 |- E. x ( ( ph -> ps ) /\ ( ps -> ch ) )
7		pm3.33 609	. 2 |- ( ( ( ph -> ps ) /\ ( ps -> ch ) ) -> ( ph -> ch ) )
8	6, 7	bnj101 30789	1 |- E. x ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 384   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

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

This theorem is referenced by:  bnj1098  30854  bnj1110  31050  bnj1118  31052  bnj1128  31058  bnj1145  31061