Theorem eqvincg 3329

Description: A variable introduction law for class equality, closed form. (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 3215	. . . 4 |- ( A e. V -> E. x x = A )
2		ax-1 6	. . . . . 6 |- ( x = A -> ( A = B -> x = A ) )
3		eqtr 2641	. . . . . . 7 |- ( ( x = A /\ A = B ) -> x = B )
4	3	ex 450	. . . . . 6 |- ( x = A -> ( A = B -> x = B ) )
5	2, 4	jca 554	. . . . 5 |- ( x = A -> ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
6	5	eximi 1762	. . . 4 |- ( E. x x = A -> E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
7		pm3.43 906	. . . . 5 |- ( ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> ( A = B -> ( x = A /\ x = B ) ) )
8	7	eximi 1762	. . . 4 |- ( E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> E. x ( A = B -> ( x = A /\ x = B ) ) )
9	1, 6, 8	3syl 18	. . 3 |- ( A e. V -> E. x ( A = B -> ( x = A /\ x = B ) ) )
10		19.37v 1910	. . 3 |- ( E. x ( A = B -> ( x = A /\ x = B ) ) <-> ( A = B -> E. x ( x = A /\ x = B ) ) )
11	9, 10	sylib 208	. 2 |- ( A e. V -> ( A = B -> E. x ( x = A /\ x = B ) ) )
12		eqtr2 2642	. . 3 |- ( ( x = A /\ x = B ) -> A = B )
13	12	exlimiv 1858	. 2 |- ( E. x ( x = A /\ x = B ) -> A = B )
14	11, 13	impbid1 215	1 |- ( A e. V -> ( A = B <-> E. x ( x = A /\ x = B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  eqvinc  3330  funcnv5mpt  29469