Theorem sbft 1769

Description: Substitution has no effect on a non-free 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 1763	. . 3 |- ( [ y / x ] ph -> E. x ph )
2		19.9t 1573	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
3	1, 2	syl5ib 152	. 2 |- ( F/ x ph -> ( [ y / x ] ph -> ph ) )
4		nfr 1451	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
5		stdpc4 1698	. . 3 |- ( A. x ph -> [ y / x ] ph )
6	4, 5	syl6 33	. 2 |- ( F/ x ph -> ( ph -> [ y / x ] ph ) )
7	3, 6	impbid 127	1 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   F/wnf 1389   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbctt  2880