Theorem ceqsex 1834

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 416	. . 3 |- ((x = A /\ ph) -> ps)
4	1, 3	19.23ai 1064	. 2 |- (E.x(x = A /\ ph) -> ps)
5		ceqsex.2	. . . 4 |- A e. V
6	5	isseti 1815	. . 3 |- E.x x = A
7	2	biimprcd 156	. . . . 5 |- (ps -> (x = A -> ph))
8	1, 7	19.21ai 998	. . . 4 |- (ps -> A.x(x = A -> ph))
9		exintr 1117	. . . 4 |- (A.x(x = A -> ph) -> (E.x x = A -> E.x(x = A /\ ph)))
10	8, 9	syl 10	. . 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 157	1 |- (E.x(x = A /\ ph) <-> ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  ceqsexv 1835  ceqsex2 1836

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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