Theorem eqvincf 1881

Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
eqvincf.1	|- (y e. A -> A.x y e. A)
eqvincf.2	|- (y e. B -> A.x y e. B)
eqvincf.3	|- A e. V

Assertion

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

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem eqvincf

Step	Hyp	Ref	Expression
1		eqvincf.3	. . 3 |- A e. V
2	1	eqvinc 1880	. 2 |- (A = B <-> E.y(y = A /\ y = B))
3		eqvincf.1	. . . . 5 |- (y e. A -> A.x y e. A)
4	3	hbeleq 1565	. . . 4 |- (y = A -> A.x y = A)
5		eqvincf.2	. . . . 5 |- (y e. B -> A.x y e. B)
6	5	hbeleq 1565	. . . 4 |- (y = B -> A.x y = B)
7	4, 6	hban 1008	. . 3 |- ((y = A /\ y = B) -> A.x(y = A /\ y = B))
8		ax-17 970	. . 3 |- ((x = A /\ x = B) -> A.y(x = A /\ x = B))
9		eqeq1 1479	. . . 4 |- (y = x -> (y = A <-> x = A))
10		eqeq1 1479	. . . 4 |- (y = x -> (y = B <-> x = B))
11	9, 10	anbi12d 627	. . 3 |- (y = x -> ((y = A /\ y = B) <-> (x = A /\ x = B)))
12	7, 8, 11	cbvex 1165	. 2 |- (E.y(y = A /\ y = B) <-> E.x(x = A /\ x = B))
13	2, 12	bitr 173	1 |- (A = B <-> E.x(x = A /\ x = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  Vcvv 1808

This theorem is referenced by:  fvopab4gf 3776  fvopabgf 3782  fvopabnf 3783

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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