Theorem sbft 1728

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 1723	. . 3 |- ( [ y / x ] ph -> E. x ph )
2		19.9t 1533	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
3	1, 2	syl5ib 143	. 2 |- ( F/ x ph -> ( [ y / x ] ph -> ph ) )
4		nfr 1411	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
5		stdpc4 1658	. . 3 |- ( A. x ph -> [ y / x ] ph )
6	4, 5	syl6 29	. 2 |- ( F/ x ph -> ( ph -> [ y / x ] ph ) )
7	3, 6	impbid 120	1 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbctt  2824