Theorem bj-sbftv 32763

Description: Version of sbft 2379 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-sbftv	|- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bj-sbftv

Step	Hyp	Ref	Expression
1		spsbe 1884	. . 3 |- ( [ y / x ] ph -> E. x ph )
2		19.9t 2071	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
3	1, 2	syl5ib 234	. 2 |- ( F/ x ph -> ( [ y / x ] ph -> ph ) )
4		nf5r 2064	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
5		bj-stdpc4v 32754	. . 3 |- ( A. x ph -> [ y / x ] ph )
6	4, 5	syl6 35	. 2 |- ( F/ x ph -> ( ph -> [ y / x ] ph ) )
7	3, 6	impbid 202	1 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708   [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-nf 1710  df-sb 1881

This theorem is referenced by:  bj-sbfv  32764