Theorem ceqsex 2324

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:   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 460	. . 3 |- ((x = A /\ ph) -> ps)
4	1, 3	19.23ai 1412	. 2 |- (E.x(x = A /\ ph) -> ps)
5		ceqsex.2	. . . 4 |- A e. _V
6	5	isseti 2297	. . 3 |- E.x x = A
7	2	biimprcd 173	. . . . 5 |- (ps -> (x = A -> ph))
8	1, 7	19.21ai 1345	. . . 4 |- (ps -> A.x(x = A -> ph))
9		exintr 1475	. . . 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 55	. 2 |- (ps -> E.x(x = A /\ ph))
12	4, 11	impbii 174	1 |- (E.x(x = A /\ ph) <-> ps)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  ceqsexv 2325  ceqsex2 2326  ceqsex2OLD 2327  bnj577 12549

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

Copyright terms: Public domain