Theorem bj-sb2v 32753

Description: Version of sb2 2352 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-sb2v	|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-sb2v

Step	Hyp	Ref	Expression
1		sp 2053	. 2 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
2		equs4v 1930	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
3		df-sb 1881	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4	1, 2, 3	sylanbrc 698	1 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   [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-12 2047

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

This theorem is referenced by:  bj-stdpc4v  32754  bj-sb3v  32756  bj-hbs1  32758  bj-hbsb2av  32760  bj-equsb1v  32762  bj-sb6  32767