Theorem equsexh 1963

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
equsexh.1	|- (ps -> A.xps)
equsexh.2	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
equsexh	|- (E.x(x = y /\ ph) <-> ps)

Proof of Theorem equsexh

Step	Hyp	Ref	Expression
1		equsexh.1	. . 3 |- (ps -> A.xps)
2	1	nfi 1551	. 2 |- F/xps
3		equsexh.2	. 2 |- (x = y -> (ph <-> ps))
4	2, 3	equsex 1962	1 |- (E.x(x = y /\ ph) <-> ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ 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-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cleljust  2014