Theorem sb2 1790

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 1533	. 2 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
2		equs4 1748	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
3		df-sb 1786	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4	1, 2, 3	sylanbrc 417	1 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   E.wex 1515   [wsb 1785

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-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-i9 1553  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-sb 1786

This theorem is referenced by:  stdpc4  1798  equsb1  1808  equsb2  1809  sbiedh  1810  sb6f  1826  hbsb2a  1829  hbsb2e  1830  sbcof2  1833  sb3  1854  sb4b  1857  sb4bor  1858  hbsb2  1859  nfsb2or  1860  sb6rf  1876  sbi1v  1915  sbalyz  2027  iota4  5251