Theorem hbsb2e 1800

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 1798	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
2		sb2 1760	. . 3 |- ( A. x ( x = y -> E. y ph ) -> [ y / x ] E. y ph )
3	2	a5i 1536	. 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 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-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-11 1499  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by: (None)