Theorem sb4 1223

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 1221	. 2 |- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))
2		sb1 1176	. 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 954   = wceq 956  E.wex 980  [wsbc 1170

This theorem is referenced by:  sb4b 1224  dfsb2 1225  hbsb2 1227  sbn 1231  sbi1 1232  hbsb4 1248  sbal1 1346

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-10 966  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-10o 1140  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

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