Theorem bj-sbfvv 32765

Description: Version of sbf 2380 with two dv conditions, which does not require ax-10 2019 nor ax-13 2246. (Contributed by BJ, 1-May-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-sbfvv	|- ( [ y / x ] ph <-> ph )

Distinct variable groups:   x, y   ph, x

Allowed substitution hint:   ph( y)

Proof of Theorem bj-sbfvv

Step	Hyp	Ref	Expression
1		spsbe 1884	. . 3 |- ( [ y / x ] ph -> E. x ph )
2		19.9v 1896	. . 3 |- ( E. x ph <-> ph )
3	1, 2	sylib 208	. 2 |- ( [ y / x ] ph -> ph )
4		ax-5 1839	. . 3 |- ( ph -> A. x ph )
5		bj-stdpc4v 32754	. . 3 |- ( A. x ph -> [ y / x ] ph )
6	4, 5	syl 17	. 2 |- ( ph -> [ y / x ] ph )
7	3, 6	impbii 199	1 |- ( [ y / x ] ph <-> ph )

Syntax hints:   <-> wb 196   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-vjust2  33015