Theorem bnj1101 30855

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

Hypotheses

Ref	Expression
bnj1101.1	|- E. x ( ph -> ps )
bnj1101.2	|- ( ch -> ph )

Assertion

Ref	Expression
bnj1101	|- E. x ( ch -> ps )

Proof of Theorem bnj1101

Step	Hyp	Ref	Expression
1		bnj1101.1	. . 3 |- E. x ( ph -> ps )
2		pm3.42 583	. . 3 |- ( ( ph -> ps ) -> ( ( ch /\ ph ) -> ps ) )
3	1, 2	bnj101 30789	. 2 |- E. x ( ( ch /\ ph ) -> ps )
4		bnj1101.2	. . . . 5 |- ( ch -> ph )
5	4	pm4.71i 664	. . . 4 |- ( ch <-> ( ch /\ ph ) )
6	5	imbi1i 339	. . 3 |- ( ( ch -> ps ) <-> ( ( ch /\ ph ) -> ps ) )
7	6	exbii 1774	. 2 |- ( E. x ( ch -> ps ) <-> E. x ( ( ch /\ ph ) -> ps ) )
8	3, 7	mpbir 221	1 |- E. x ( ch -> ps )

Syntax hints:   -> wi 4   /\ wa 384   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:  bnj1110  31050  bnj1128  31058  bnj1145  31061