Theorem bj-sb3v 32756

Description: Version of sb3 2355 with a dv condition, which does not require ax-13 2246. This allows to remove ax-13 2246 from sb5 2430 (see bj-sb5 32768). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-sb3v	|- ( E. x ( x = y /\ ph ) -> [ y / x ] ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-sb3v

Step	Hyp	Ref	Expression
1		sb56 2150	. 2 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
2		bj-sb2v 32753	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	1, 2	sylbi 207	1 |- ( E. 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-10 2019  ax-12 2047

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: (None)