Theorem cleqh 2572

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2646. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.)

Hypotheses

Ref	Expression
cleqh.1
cleqh.2

Assertion

Ref	Expression
cleqh

Distinct variable groups:   ,   ,   ,

Proof of Theorem cleqh

Step	Hyp	Ref	Expression
1		dfcleq 2450	. 2
2		nfv 1707	. . 3
3		cleqh.1	. . . . 5
4	3	nfi 1623	. . . 4
5		cleqh.2	. . . . 5
6	5	nfi 1623	. . . 4
7	4, 6	nfbi 1934	. . 3
8		eleq1 2529	. . . 4
9		eleq1 2529	. . . 4
10	8, 9	bibi12d 321	. . 3
11	2, 7, 10	cbval 2021	. 2
12	1, 11	bitr4i 252	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  e.wcel 1818

This theorem is referenced by:  abeq2  2581  abbi  2588  cleqf  2646  abeq2f  27398

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-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-cleq 2449  df-clel 2452