Theorem hbsbv 1970

Description: This is a version of hbsb 1978 with an extra distinct variable constraint, on z and x. (Contributed by Jim Kingdon, 25-Dec-2017.)

Hypothesis

Ref	Expression
hbsbv.1	|- ( ph -> A. z ph )

Assertion

Ref	Expression
hbsbv	|- ( [ y / x ] ph -> A. z [ y / x ] ph )

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem hbsbv

Step	Hyp	Ref	Expression
1		df-sb 1787	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2		ax-17 1550	. . . 4 |- ( x = y -> A. z x = y )
3		hbsbv.1	. . . 4 |- ( ph -> A. z ph )
4	2, 3	hbim 1569	. . 3 |- ( ( x = y -> ph ) -> A. z ( x = y -> ph ) )
5	2, 3	hban 1571	. . . 4 |- ( ( x = y /\ ph ) -> A. z ( x = y /\ ph ) )
6	5	hbex 1660	. . 3 |- ( E. x ( x = y /\ ph ) -> A. z E. x ( x = y /\ ph ) )
7	4, 6	hban 1571	. 2 |- ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> A. z ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
8	1, 7	hbxfrbi 1496	1 |- ( [ y / x ] ph -> A. z [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   E.wex 1516   [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-sb 1787

This theorem is referenced by:  sbco2vlem  1973  2sb5rf  2018  2sb6rf  2019