Theorem equs5e 1888

Description: A property related to substitution that unlike equs5 1996 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
equs5e	|- (E.x(x = y /\ ph) -> A.x(x = y -> E.yph))

Proof of Theorem equs5e

Step	Hyp	Ref	Expression
1		nfe1 1732	. 2 |- F/xE.x(x = y /\ ph)
2		equs3 1644	. . 3 |- (E.x(x = y /\ ph) <-> -. A.x(x = y -> -. ph))
3		ax-11 1746	. . . . 5 |- (x = y -> (A.y -. ph -> A.x(x = y -> -. ph)))
4	3	con3rr3 128	. . . 4 |- (-. A.x(x = y -> -. ph) -> (x = y -> -. A.y -. ph))
5		df-ex 1542	. . . 4 |- (E.yph <-> -. A.y -. ph)
6	4, 5	syl6ibr 218	. . 3 |- (-. A.x(x = y -> -. ph) -> (x = y -> E.yph))
7	2, 6	sylbi 187	. 2 |- (E.x(x = y /\ ph) -> (x = y -> E.yph))
8	1, 7	alrimi 1765	1 |- (E.x(x = y /\ ph) -> A.x(x = y -> E.yph))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  sb4e  1924