Theorem sb4 1825

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 1759	. 2 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		equs5 1822	. 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 103   A.wal 1346   E.wex 1485   [wsb 1755

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756

This theorem is referenced by:  sb4b  1827  hbsb2  1829