Theorem bnj982 33980

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

Hypotheses

Ref	Expression
bnj982.1	|- ( ph -> A. x ph )
bnj982.2	|- ( ps -> A. x ps )
bnj982.3	|- ( ch -> A. x ch )
bnj982.4	|- ( th -> A. x th )

Assertion

Ref	Expression
bnj982	|- ( ( ph /\ ps /\ ch /\ th ) -> A. x ( ph /\ ps /\ ch /\ th ) )

Proof of Theorem bnj982

Step	Hyp	Ref	Expression
1		df-bnj17 33882	. 2 |- ( ( ph /\ ps /\ ch /\ th ) <-> ( ( ph /\ ps /\ ch ) /\ th ) )
2		bnj982.1	. . . 4 |- ( ph -> A. x ph )
3		bnj982.2	. . . 4 |- ( ps -> A. x ps )
4		bnj982.3	. . . 4 |- ( ch -> A. x ch )
5	2, 3, 4	hb3an 1933	. . 3 |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )
6		bnj982.4	. . 3 |- ( th -> A. x th )
7	5, 6	hban 1932	. 2 |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> A. x ( ( ph /\ ps /\ ch ) /\ th ) )
8	1, 7	hbxfrbi 1644	1 |- ( ( ph /\ ps /\ ch /\ th ) -> A. x ( ph /\ ps /\ ch /\ th ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   A.wal 1393   /\ w-bnj17 33881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-12 1855

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1614  df-nf 1618  df-bnj17 33882

This theorem is referenced by:  bnj1096  33984  bnj1311  34223  bnj1445  34243