Theorem eqrdav 2455

Description: Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)

Hypotheses

Ref	Expression
eqrdav.1
eqrdav.2
eqrdav.3

Assertion

Ref	Expression
eqrdav

Distinct variable groups:   ,   ,   ,

Proof of Theorem eqrdav

Step	Hyp	Ref	Expression
1		eqrdav.1	. . . 4
2		eqrdav.3	. . . . . 6
3	2	biimpd 207	. . . . 5
4	3	impancom 440	. . . 4
5	1, 4	mpd 15	. . 3
6		eqrdav.2	. . . 4
7	2	biimprd 223	. . . . 5
8	7	impancom 440	. . . 4
9	6, 8	mpd 15	. . 3
10	5, 9	impbida 832	. 2
11	10	eqrdv 2454	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818

This theorem is referenced by:  boxcutc  7532  supminf  11198  f1omvdconj  16471  fmucndlem  20794  ballotlemsima  28454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-cleq 2449