Theorem sbft 1859

Description: Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)

Assertion

Ref	Expression
sbft	|- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )

Proof of Theorem sbft

Step	Hyp	Ref	Expression
1		spsbe 1853	. . 3 |- ( [ y / x ] ph -> E. x ph )
2		19.9t 1653	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
3	1, 2	imbitrid 154	. 2 |- ( F/ x ph -> ( [ y / x ] ph -> ph ) )
4		nfr 1529	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
5		stdpc4 1786	. . 3 |- ( A. x ph -> [ y / x ] ph )
6	4, 5	syl6 33	. 2 |- ( F/ x ph -> ( ph -> [ y / x ] ph ) )
7	3, 6	impbid 129	1 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   F/wnf 1471   E.wex 1503   [wsb 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sbctt  3044