Theorem bj-hbsb2av 32760

Description: Version of hbsb2a 2361 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-hbsb2av	|- ( [ y / x ] A. y ph -> A. x [ y / x ] ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-hbsb2av

Step	Hyp	Ref	Expression
1		sb4a 2357	. 2 |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) )
2		bj-sb2v 32753	. . 3 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	2	axc4i 2131	. 2 |- ( A. x ( x = y -> ph ) -> A. x [ y / x ] ph )
4	1, 3	syl 17	1 |- ( [ y / x ] A. y ph -> A. x [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1481   [wsb 1880

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  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  bj-hbsb3v  32761