Theorem hbsb2e 2153

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 2051	. 2 |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )
2		sb2 2144	. . 3 |- ( A. x ( x = y -> E. y ph ) -> [ y / x ] E. y ph )
3	2	axc4i 1952	. 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 1435   E.wex 1659   [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-12 1904  ax-13 2052

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787

This theorem is referenced by: (None)