Theorem ceqsex 1370

Description: Elimination of an existential quantifier, using implicit substitution.

Hypotheses

Ref	Expression
ceqsex.1	|- (ps -> A.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(s):   x,A

Proof of Theorem ceqsex

Step	Hyp	Ref	Expression
1		ceqsex.1	. . 3 |- (ps -> A.xps)
2		ceqsex.3	. . . 4 |- (x = A -> (ph <-> ps))
3	2	biimpa 324	. . 3 |- ((x = A /\ ph) -> ps)
4	1, 3	19.23ai 746	. 2 |- (E.x(x = A /\ ph) -> ps)
5		ceqsex.2	. . . 4 |- A e. V
6	5	isseti 1352	. . 3 |- E.x x = A
7	2	biimprcd 138	. . . . 5 |- (ps -> (x = A -> ph))
8	1, 7	19.21ai 740	. . . 4 |- (ps -> A.x(x = A -> ph))
9		exintr 793	. . . 4 |- (A.x(x = A -> ph) -> (E.x x = A -> E.x(x = A /\ ph)))
10	8, 9	syl 12	. . 3 |- (ps -> (E.x x = A -> E.x(x = A /\ ph)))
11	6, 10	mpi 44	. 2 |- (ps -> E.x(x = A /\ ph))
12	4, 11	impbi 139	1 |- (E.x(x = A /\ ph) <-> ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  ceqsexv 1371

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

metamath.org