Theorem hbsb2a 1829

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

Assertion

Ref	Expression
hbsb2a	|- ( [ y / x ] A. y ph -> A. x [ y / x ] ph )

Proof of Theorem hbsb2a

Step	Hyp	Ref	Expression
1		sb4a 1824	. 2 |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) )
2		sb2 1790	. . 3 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	2	a5i 1566	. 2 |- ( A. x ( x = y -> ph ) -> A. x [ y / x ] ph )
4	1, 3	syl 14	1 |- ( [ y / x ] A. y ph -> A. x [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1371   [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-11 1529  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:  hbsb3  1831