Theorem sb4b 2207

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 2205	. 2 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
2		sb2 2201	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	1, 2	impbid1 208	1 |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   A.wal 1450   [wsb 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806

This theorem is referenced by:  sbcom3  2260  sbal1  2309  sbal2  2310  wl-sb6nae  31956  wl-sbalnae  31962  wl-sbcom3  31989