Theorem bj-sbnf 32828

Description: Move non-free predicate in and out of substitution; see sbal 2462 and sbex 2463. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-sbnf	|- ( [ z / y ] F/ x ph <-> F/ x [ z / y ] ph )

Distinct variable groups:   x, y   x, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem bj-sbnf

Step	Hyp	Ref	Expression
1		sbim 2395	. . . 4 |- ( [ z / y ] ( ph -> A. x ph ) <-> ( [ z / y ] ph -> [ z / y ] A. x ph ) )
2		sbal 2462	. . . . 5 |- ( [ z / y ] A. x ph <-> A. x [ z / y ] ph )
3	2	imbi2i 326	. . . 4 |- ( ( [ z / y ] ph -> [ z / y ] A. x ph ) <-> ( [ z / y ] ph -> A. x [ z / y ] ph ) )
4	1, 3	bitri 264	. . 3 |- ( [ z / y ] ( ph -> A. x ph ) <-> ( [ z / y ] ph -> A. x [ z / y ] ph ) )
5	4	albii 1747	. 2 |- ( A. x [ z / y ] ( ph -> A. x ph ) <-> A. x ( [ z / y ] ph -> A. x [ z / y ] ph ) )
6		nf5 2116	. . . 4 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
7	6	sbbii 1887	. . 3 |- ( [ z / y ] F/ x ph <-> [ z / y ] A. x ( ph -> A. x ph ) )
8		sbal 2462	. . 3 |- ( [ z / y ] A. x ( ph -> A. x ph ) <-> A. x [ z / y ] ( ph -> A. x ph ) )
9	7, 8	bitri 264	. 2 |- ( [ z / y ] F/ x ph <-> A. x [ z / y ] ( ph -> A. x ph ) )
10		nf5 2116	. 2 |- ( F/ x [ z / y ] ph <-> A. x ( [ z / y ] ph -> A. x [ z / y ] ph ) )
11	5, 9, 10	3bitr4i 292	1 |- ( [ z / y ] F/ x ph <-> F/ x [ z / y ] ph )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  bj-nfcf  32920