Theorem bnj1095 29167

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

Hypothesis

Ref	Expression
bnj1095.1	|- ( ph <-> A. x e. A ps )

Assertion

Ref	Expression
bnj1095	|- ( ph -> A. x ph )

Proof of Theorem bnj1095

Step	Hyp	Ref	Expression
1		bnj1095.1	. 2 |- ( ph <-> A. x e. A ps )
2		hbra1 2786	. 2 |- ( A. x e. A ps -> A. x A. x e. A ps )
3	1, 2	hbxfrbi 1664	1 |- ( ph -> A. x ph )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1403   A.wral 2754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878

This theorem depends on definitions:  df-bi 185  df-ex 1634  df-nf 1638  df-ral 2759

This theorem is referenced by:  bnj1379  29216  bnj605  29292  bnj594  29297  bnj607  29301  bnj911  29317  bnj964  29328  bnj983  29336  bnj1093  29363  bnj1123  29369  bnj1145  29376  bnj1417  29424