Theorem sbft 1871

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 1865	. . 3 |- ( [ y / x ] ph -> E. x ph )
2		19.9t 1665	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
3	1, 2	imbitrid 154	. 2 |- ( F/ x ph -> ( [ y / x ] ph -> ph ) )
4		nfr 1541	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
5		stdpc4 1798	. . 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 1371   F/wnf 1483   E.wex 1515   [wsb 1785

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  sbctt  3065