Theorem sb4 1880

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

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		sb1 1814	. 2 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		equs5 1877	. 2 |- ( -. A. x x = y -> ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
3	1, 2	syl5 32	1 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1396   E.wex 1541   [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811

This theorem is referenced by:  sb4b  1882  hbsb2  1884