Theorem sb4b 2358

Description: Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)

Assertion

Ref	Expression
sb4b	|- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Proof of Theorem sb4b

Step	Hyp	Ref	Expression
1		sb4 2356	. 2 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
2		sb2 2352	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	1, 2	impbid1 215	1 |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   [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:  sbcom3  2411  sbal1  2460  sbal2  2461  wl-sb6nae  33339  wl-sbalnae  33345  wl-sbcom3  33372