Theorem eqvincg 2668

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 2568	. . . 4 |- ( A e. V -> E. x x = A )
2		ax-1 5	. . . . . 6 |- ( x = A -> ( A = B -> x = A ) )
3		eqtr 2057	. . . . . . 7 |- ( ( x = A /\ A = B ) -> x = B )
4	3	ex 108	. . . . . 6 |- ( x = A -> ( A = B -> x = B ) )
5	2, 4	jca 290	. . . . 5 |- ( x = A -> ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
6	5	eximi 1491	. . . 4 |- ( E. x x = A -> E. x ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) )
7		pm3.43 534	. . . . 5 |- ( ( ( A = B -> x = A ) /\ ( A = B -> x = B ) ) -> ( A = B -> ( x = A /\ x = B ) ) )
8	7	eximi 1491	. . . 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 1421	. . . 4 |- F/ x A = B
11	10	19.37-1 1564	. . 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 2058	. . 3 |- ( ( x = A /\ x = B ) -> A = B )
14	13	exlimiv 1489	. 2 |- ( E. x ( x = A /\ x = B ) -> A = B )
15	12, 14	impbid1 130	1 |- ( A e. V -> ( A = B <-> E. x ( x = A /\ x = B ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  dff13  5407  f1eqcocnv  5431