Theorem hbsb2e 1728

Description: Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
hbsb2e	|- ( [ y / x ] ph -> A. x [ y / x ] E. y ph )

Proof of Theorem hbsb2e

Step	Hyp	Ref	Expression
1		sb4e 1726	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
2		sb2 1690	. . 3 |- ( A. x ( x = y -> E. y ph ) -> [ y / x ] E. y ph )
3	2	a5i 1475	. 2 |- ( A. x ( x = y -> E. y ph ) -> A. x [ y / x ] E. y ph )
4	1, 3	syl 14	1 |- ( [ y / x ] ph -> A. x [ y / x ] E. y ph )

Syntax hints:   -> wi 4   A.wal 1282   E.wex 1421   [wsb 1685

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-11 1437  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by: (None)