Theorem wl-equsb4 33338

Description: Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019.)

Assertion

Ref	Expression
wl-equsb4	|- ( -. A. x x = z -> ( [ y / x ] y = z <-> y = z ) )

Proof of Theorem wl-equsb4

Step	Hyp	Ref	Expression
1		nfeqf 2301	. . . 4 |- ( ( -. A. x x = y /\ -. A. x x = z ) -> F/ x y = z )
2	1	ex 450	. . 3 |- ( -. A. x x = y -> ( -. A. x x = z -> F/ x y = z ) )
3		sbft 2379	. . 3 |- ( F/ x y = z -> ( [ y / x ] y = z <-> y = z ) )
4	2, 3	syl6com 37	. 2 |- ( -. A. x x = z -> ( -. A. x x = y -> ( [ y / x ] y = z <-> y = z ) ) )
5		sbequ12r 2112	. . . 4 |- ( y = x -> ( [ y / x ] y = z <-> y = z ) )
6	5	equcoms 1947	. . 3 |- ( x = y -> ( [ y / x ] y = z <-> y = z ) )
7	6	sps 2055	. 2 |- ( A. x x = y -> ( [ y / x ] y = z <-> y = z ) )
8	4, 7	pm2.61d2 172	1 |- ( -. A. x x = z -> ( [ y / x ] y = z <-> y = z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   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-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)