Theorem eqvincg 2763

Description: A variable introduction law for class equality, deduction version. (Contributed by Thierry Arnoux, 2-Mar-2017.)

Assertion

Ref	Expression
eqvincg	|- ( A e. V -> ( A = B <-> E. x ( x = A /\ x = B ) ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem eqvincg

Step	Hyp	Ref	Expression
1		elisset 2655	. . . 4 |- ( A e. V -> E. x x = A )
2		ax-1 5	. . . . . 6 |- ( x = A -> ( A = B -> x = A ) )
3		eqtr 2117	. . . . . . 7 |- ( ( x = A /\ A = B ) -> x = B )
4	3	ex 114	. . . . . 6 |- ( x = A -> ( A = B -> x = B ) )
5	2, 4	jca 302	. . . . 5 |- ( x = A -> ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
6	5	eximi 1547	. . . 4 |- ( E. x x = A -> E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
7		pm3.43 572	. . . . 5 |- ( ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> ( A = B -> ( x = A /\ x = B ) ) )
8	7	eximi 1547	. . . 4 |- ( E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> E. x ( A = B -> ( x = A /\ x = B ) ) )
9	1, 6, 8	3syl 17	. . 3 |- ( A e. V -> E. x ( A = B -> ( x = A /\ x = B ) ) )
10		nfv 1476	. . . 4 |- F/ x A = B
11	10	19.37-1 1620	. . 3 |- ( E. x ( A = B -> ( x = A /\ x = B ) ) -> ( A = B -> E. x ( x = A /\ x = B ) ) )
12	9, 11	syl 14	. 2 |- ( A e. V -> ( A = B -> E. x ( x = A /\ x = B ) ) )
13		eqtr2 2118	. . 3 |- ( ( x = A /\ x = B ) -> A = B )
14	13	exlimiv 1545	. 2 |- ( E. x ( x = A /\ x = B ) -> A = B )
15	12, 14	impbid1 141	1 |- ( A e. V -> ( A = B <-> E. x ( x = A /\ x = B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   E.wex 1436   e. wcel 1448

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-v 2643

This theorem is referenced by:  dff13  5601  f1eqcocnv  5624