Theorem hbsb2 2359

Description: Bound-variable hypothesis builder for substitution. (Contributed by NM, 14-May-1993.)

Assertion

Ref	Expression
hbsb2	|- ( -. A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) )

Proof of Theorem hbsb2

Step	Hyp	Ref	Expression
1		sb4 2356	. 2 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
2		sb2 2352	. . 3 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	2	axc4i 2131	. 2 |- ( A. x ( x = y -> ph ) -> A. x [ y / x ] ph )
4	1, 3	syl6 35	1 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  nfsb2  2360  hbs1  2436