Theorem ceqsex 2894

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

Hypotheses

Ref	Expression
ceqsex.1	|- F/xps
ceqsex.2	|- A e. _V
ceqsex.3	|- (x = A -> (ph <-> ps))

Assertion

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

Distinct variable group:   x,A

Allowed substitution hints:   ph(x)   ps(x)

Proof of Theorem ceqsex

Step	Hyp	Ref	Expression
1		ceqsex.1	. . 3 |- F/xps
2		ceqsex.3	. . . 4 |- (x = A -> (ph <-> ps))
3	2	biimpa 470	. . 3 |- ((x = A /\ ph) -> ps)
4	1, 3	exlimi 1803	. 2 |- (E.x(x = A /\ ph) -> ps)
5	2	biimprcd 216	. . . 4 |- (ps -> (x = A -> ph))
6	1, 5	alrimi 1765	. . 3 |- (ps -> A.x(x = A -> ph))
7		ceqsex.2	. . . 4 |- A e. _V
8	7	isseti 2866	. . 3 |- E.x x = A
9		exintr 1614	. . 3 |- (A.x(x = A -> ph) -> (E.x x = A -> E.x(x = A /\ ph)))
10	6, 8, 9	ee10 1376	. 2 |- (ps -> E.x(x = A /\ ph))
11	4, 10	impbii 180	1 |- (E.x(x = A /\ ph) <-> ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710  _Vcvv 2860

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  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  ceqsexv  2895  ceqsex2  2896