Theorem sbft 2379

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 1884	. . 3 |- ( [ y / x ] ph -> E. x ph )
2		19.9t 2071	. . 3 |- ( F/ x ph -> ( E. x ph <-> ph ) )
3	1, 2	syl5ib 234	. 2 |- ( F/ x ph -> ( [ y / x ] ph -> ph ) )
4		nf5r 2064	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
5		stdpc4 2353	. . 3 |- ( A. x ph -> [ y / x ] ph )
6	4, 5	syl6 35	. 2 |- ( F/ x ph -> ( ph -> [ y / x ] ph ) )
7	3, 6	impbid 202	1 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   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-12 2047  ax-13 2246

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

This theorem is referenced by:  sbf  2380  sbctt  3500  wl-sbrimt  33331  wl-sblimt  33332  wl-equsb4  33338