Theorem hbsb2a 1817

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 1812	. 2 |- ( [ y / x ] A. y ph -> A. x ( x = y -> ph ) )
2		sb2 1778	. . 3 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	2	a5i 1554	. 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 1362   [wsb 1773

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-11 1517  ax-4 1521  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-sb 1774

This theorem is referenced by:  hbsb3  1819