Theorem bj-sb3b 32804

Description: Simplified definition of substitution when variables are distinct. This is to sb3 2355 what sb4b 2358 is to sb4 2356. Actually, one may keep only bj-sb3b 32804 and sb4b 2358 in the database, renaming them sb3 and sb4. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-sb3b	|- ( -. A. x x = y -> ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) )

Proof of Theorem bj-sb3b

Step	Hyp	Ref	Expression
1		sb1 1883	. 2 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		sb3 2355	. 2 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) )
3	1, 2	impbid2 216	1 |- ( -. A. x x = y -> ( [ y / x ] ph <-> E. x ( x = y /\ ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   [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-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)