Theorem nfeq 2383

Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfnfc.1	|- F/_ x A
nfeq.2	|- F/_ x B

Assertion

Ref	Expression
nfeq	|- F/ x A = B

Proof of Theorem nfeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2225	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2369	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2369	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1638	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1625	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1523	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1396   = wceq 1398   F/wnf 1509   e. wcel 2202   F/_wnfc 2362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2364

This theorem is referenced by:  nfel  2384  nfeq1  2385  nfeq2  2387  nfne  2496  raleqf  2727  rexeqf  2728  reueq1f  2729  rmoeq1f  2730  rabeqf  2793  sbceqg  3144  csbhypf  3167  nfiotadw  5296  nffn  5433  nffo  5567  fvmptdf  5743  mpteqb  5746  fvmptf  5748  eqfnfv2f  5757  dff13f  5921  ovmpos  6155  ov2gf  6156  ovmpodxf  6157  ovmpodf  6163  eqerlem  6776  sumeq2  11999  fsumadd  12047  prodeq1f  12193  prodeq2  12198  txcnp  15082  cnmpt11  15094  cnmpt21  15102  cnmptcom  15109  dvmptfsum  15536  lgseisenlem2  15890