Theorem sb4 1221

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 1219	. 2 |- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))
2		sb1 1174	. 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 952   = wceq 954  E.wex 978  [wsbc 1168

This theorem is referenced by:  sb4b 1222  dfsb2 1223  hbsb2 1225  sbn 1229  sbi1 1230  hbsb4 1246  sbal1 1344

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170

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