Theorem equs5a 1887

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
equs5a	|- (E.x(x = y /\ A.yph) -> A.x(x = y -> ph))

Proof of Theorem equs5a

Step	Hyp	Ref	Expression
1		nfa1 1788	. 2 |- F/xA.x(x = y -> ph)
2		ax-11 1746	. . 3 |- (x = y -> (A.yph -> A.x(x = y -> ph)))
3	2	imp 418	. 2 |- ((x = y /\ A.yph) -> A.x(x = y -> ph))
4	1, 3	exlimi 1803	1 |- (E.x(x = y /\ A.yph) -> A.x(x = y -> ph))

Syntax hints:   -> 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:  sb4a  1923  equs45f  1989