Theorem sb3 2355

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 2351	. 2 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) ) )
2		sb2 2352	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	1, 2	syl6bi 243	1 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  bj-sb3b  32804  wl-sb5nae  33340