Theorem sb4b 2100

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 2099	. 2 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
2		sb2 2095	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	1, 2	impbid1 203	1 |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   A.wal 1396   [wsb 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004

This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745

This theorem is referenced by:  sbcom3  2155  sbal1  2206  sbal2  2207  wl-sb6nae  30246  wl-sbalnae  30252  wl-sbcom3  30275