Theorem sb3 1759

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 1757	. 2 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
2		sb2 1697	. 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 102   A.wal 1287   E.wex 1426   [wsb 1692

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693

This theorem is referenced by: (None)