Theorem cleqh 2724

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2790. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2019.) Remove dependency on ax-13 2246. (Revised by BJ, 30-Nov-2020.)

Hypotheses

Ref	Expression
cleqh.1	|- ( y e. A -> A. x y e. A )
cleqh.2	|- ( y e. B -> A. x y e. B )

Assertion

Ref	Expression
cleqh	|- ( A = B <-> A. x ( x e. A <-> x e. B ) )

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

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem cleqh

Step	Hyp	Ref	Expression
1		dfcleq 2616	. 2 |- ( A = B <-> A. y ( y e. A <-> y e. B ) )
2		nfv 1843	. . 3 |- F/ y ( x e. A <-> x e. B )
3		cleqh.1	. . . . 5 |- ( y e. A -> A. x y e. A )
4	3	nf5i 2024	. . . 4 |- F/ x y e. A
5		cleqh.2	. . . . 5 |- ( y e. B -> A. x y e. B )
6	5	nf5i 2024	. . . 4 |- F/ x y e. B
7	4, 6	nfbi 1833	. . 3 |- F/ x ( y e. A <-> y e. B )
8		eleq1 2689	. . . 4 |- ( x = y -> ( x e. A <-> y e. A ) )
9		eleq1 2689	. . . 4 |- ( x = y -> ( x e. B <-> y e. B ) )
10	8, 9	bibi12d 335	. . 3 |- ( x = y -> ( ( x e. A <-> x e. B ) <-> ( y e. A <-> y e. B ) ) )
11	2, 7, 10	cbvalv1 2175	. 2 |- ( A. x ( x e. A <-> x e. B ) <-> A. y ( y e. A <-> y e. B ) )
12	1, 11	bitr4i 267	1 |- ( A = B <-> A. x ( x e. A <-> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618

This theorem is referenced by:  abeq2  2732  abbi  2737  cleqf  2790  abeq2f  2792  bj-abeq2  32773