Theorem sb2 1691

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

Assertion

Ref	Expression
sb2	|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )

Proof of Theorem sb2

Step	Hyp	Ref	Expression
1		ax-4 1441	. 2 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
2		equs4 1654	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
3		df-sb 1687	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4	1, 2, 3	sylanbrc 408	1 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1283   E.wex 1422   [wsb 1686

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-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-sb 1687

This theorem is referenced by:  stdpc4  1699  equsb1  1709  equsb2  1710  sbiedh  1711  sb6f  1725  hbsb2a  1728  hbsb2e  1729  sbcof2  1732  sb3  1753  sb4b  1756  sb4bor  1757  hbsb2  1758  nfsb2or  1759  sb6rf  1775  sbi1v  1813  sbalyz  1917  iota4  4915