Theorem hbsb2e 2191

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 2087	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
2		sb2 2182	. . 3 |- ( A. x ( x = y -> E. y ph ) -> [ y / x ] E. y ph )
3	2	axc4i 1979	. 2 |- ( A. x ( x = y -> E. y ph ) -> A. x [ y / x ] E. y ph )
4	1, 3	syl 17	1 |- ( [ y / x ] ph -> A. x [ y / x ] E. y ph )

Syntax hints:   -> wi 4   A.wal 1441   E.wex 1662   [wsb 1796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-12 1932  ax-13 2090

This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1663  df-nf 1667  df-sb 1797

This theorem is referenced by: (None)