Theorem sb3 1804

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 1802	. 2 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
2		sb2 1741	. 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 1330   E.wex 1469   [wsb 1736

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737

This theorem is referenced by: (None)