Theorem bnj1096 30853

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

Hypotheses

Ref	Expression
bnj1096.1	|- ( ph -> A. x ph )
bnj1096.2	|- ( ps <-> ( ch /\ th /\ ta /\ ph ) )

Assertion

Ref	Expression
bnj1096	|- ( ps -> A. x ps )

Distinct variable groups:   ch, x   ta, x   th, x

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem bnj1096

Step	Hyp	Ref	Expression
1		bnj1096.2	. 2 |- ( ps <-> ( ch /\ th /\ ta /\ ph ) )
2		ax-5 1839	. . 3 |- ( ch -> A. x ch )
3		ax-5 1839	. . 3 |- ( th -> A. x th )
4		ax-5 1839	. . 3 |- ( ta -> A. x ta )
5		bnj1096.1	. . 3 |- ( ph -> A. x ph )
6	2, 3, 4, 5	bnj982 30849	. 2 |- ( ( ch /\ th /\ ta /\ ph ) -> A. x ( ch /\ th /\ ta /\ ph ) )
7	1, 6	hbxfrbi 1752	1 |- ( ps -> A. x ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   /\ w-bnj17 30752

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-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-bnj17 30753

This theorem is referenced by:  bnj964  31013  bnj981  31020  bnj983  31021  bnj1093  31048  bnj1145  31061