Theorem hbsb2e 1730

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 1728	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
2		sb2 1692	. . 3 |- ( A. x ( x = y -> E. y ph ) -> [ y / x ] E. y ph )
3	2	a5i 1476	. 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 1283   E.wex 1422   [wsb 1687

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-11 1438  ax-4 1441  ax-i9 1464  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-sb 1688

This theorem is referenced by: (None)