Theorem eqvincf 1884

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 1883	. 2 |- (A = B <-> E.y(y = A /\ y = B))
3		eqvincf.1	. . . . 5 |- (y e. A -> A.x y e. A)
4	3	hbeleq 1567	. . . 4 |- (y = A -> A.x y = A)
5		eqvincf.2	. . . . 5 |- (y e. B -> A.x y e. B)
6	5	hbeleq 1567	. . . 4 |- (y = B -> A.x y = B)
7	4, 6	hban 1009	. . 3 |- ((y = A /\ y = B) -> A.x(y = A /\ y = B))
8		ax-17 971	. . 3 |- ((x = A /\ x = B) -> A.y(x = A /\ x = B))
9		eqeq1 1481	. . . 4 |- (y = x -> (y = A <-> x = A))
10		eqeq1 1481	. . . 4 |- (y = x -> (y = B <-> x = B))
11	9, 10	anbi12d 628	. . 3 |- (y = x -> ((y = A /\ y = B) <-> (x = A /\ x = B)))
12	7, 8, 11	cbvex 1166	. 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 954   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  fvopab4gf 3781  fvopabgf 3787  fvopabnf 3788

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

Copyright terms: Public domain