Theorem gen31 38846

Description: Virtual deduction generalizing rule for one quantifying variable and three virtual hypothesis. gen31 38846 is ggen31 38760 with virtual deductions. (Contributed by Alan Sare, 22-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
gen31.1	|- (. ph ,. ps ,. ch ->. th ).

Assertion

Ref	Expression
gen31	|- (. ph ,. ps ,. ch ->. A. x th ).

Distinct variable groups:   ch, x   ph, x   ps, x

Allowed substitution hint:   th( x)

Proof of Theorem gen31

Step	Hyp	Ref	Expression
1		gen31.1	. . . 4 |- (. ph ,. ps ,. ch ->. th ).
2	1	dfvd3i 38808	. . 3 |- ( ph -> ( ps -> ( ch -> th ) ) )
3	2	ggen31 38760	. 2 |- ( ph -> ( ps -> ( ch -> A. x th ) ) )
4	3	dfvd3ir 38809	1 |- (. ph ,. ps ,. ch ->. A. x th ).

Syntax hints:   A.wal 1481   (.wvd3 38803

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-vd3 38806

This theorem is referenced by:  onfrALTlem2VD  39125