Theorem bnj982 30849

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 30753	. 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 2129	. . 3 |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )
6		bnj982.4	. . 3 |- ( th -> A. x th )
7	5, 6	hban 2128	. 2 |- ( ( ( ph /\ ps /\ ch ) /\ th ) -> A. x ( ( ph /\ ps /\ ch ) /\ th ) )
8	1, 7	hbxfrbi 1752	1 |- ( ( ph /\ ps /\ ch /\ th ) -> A. x ( ph /\ ps /\ ch /\ th ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   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:  bnj1096  30853  bnj1311  31092  bnj1445  31112