Theorem ceqsex 3145

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
ceqsex.2
ceqsex.3

Assertion

Ref	Expression
ceqsex

Distinct variable group:   ,

Proof of Theorem ceqsex

Step	Hyp	Ref	Expression
1		ceqsex.1	. . 3
2		ceqsex.3	. . . 4
3	2	biimpa 484	. . 3
4	1, 3	exlimi 1912	. 2
5	2	biimprcd 225	. . . 4
6	1, 5	alrimi 1877	. . 3
7		ceqsex.2	. . . 4
8	7	isseti 3115	. . 3
9		exintr 1702	. . 3
10	6, 8, 9	mpisyl 18	. 2
11	4, 10	impbii 188	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  F/wnf 1616  e.wcel 1818  cvv 3109

This theorem is referenced by:  ceqsexv  3146  ceqsex2  3147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-12 1854  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111