Theorem sb4 1227

Description: One direction of a simplified definition of substitution when variables are distinct.

Assertion

Ref	Expression
sb4	|- (-. A.x x = y -> ([y / x]ph -> A.x(x = y -> ph)))

Proof of Theorem sb4

Step	Hyp	Ref	Expression
1		equs5 1225	. 2 |- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))
2		sb1 1180	. 2 |- ([y / x]ph -> E.x(x = y /\ ph))
3	1, 2	syl5 21	1 |- (-. A.x x = y -> ([y / x]ph -> A.x(x = y -> ph)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 958   = wceq 960  E.wex 984  [wsbc 1174

This theorem is referenced by:  sb4b 1228  dfsb2 1229  hbsb2 1231  sbn 1235  sbi1 1236  hbsb4 1252  sbal1 1350

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-10 970  ax-12 972  ax-4 977  ax-5o 979  ax-6o 982  ax-10o 1144  ax-11o 1222

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1176

Copyright terms: Public domain