Theorem bj-sb6 32767

Description: Remove dependency on ax-13 2246 from sb6 2429. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-sb6	|- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-sb6

Step	Hyp	Ref	Expression
1		sb1 1883	. . 3 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sb56 2150	. . 3 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )
3	1, 2	sylib 208	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> ph ) )
4		bj-sb2v 32753	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
5	3, 4	impbii 199	1 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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:  bj-sb5  32768