Theorem sb3 1803

Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sb3	|- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) )

Proof of Theorem sb3

Step	Hyp	Ref	Expression
1		equs5 1801	. 2 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
2		sb2 1740	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	1, 2	syl6 33	1 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1329   E.wex 1468   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736

This theorem is referenced by: (None)