Theorem bnj31 30785

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

Hypotheses

Ref	Expression
bnj31.1	|- ( ph -> E. x e. A ps )
bnj31.2	|- ( ps -> ch )

Assertion

Ref	Expression
bnj31	|- ( ph -> E. x e. A ch )

Proof of Theorem bnj31

Step	Hyp	Ref	Expression
1		bnj31.1	. 2 |- ( ph -> E. x e. A ps )
2		bnj31.2	. . 3 |- ( ps -> ch )
3	2	reximi 3011	. 2 |- ( E. x e. A ps -> E. x e. A ch )
4	1, 3	syl 17	1 |- ( ph -> E. x e. A ch )

Syntax hints:   -> wi 4   E.wrex 2913

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  df-ral 2917  df-rex 2918

This theorem is referenced by:  bnj168  30798  bnj110  30928  bnj906  31000  bnj1253  31085  bnj1280  31088  bnj1296  31089  bnj1371  31097  bnj1497  31128  bnj1498  31129  bnj1501  31135