Theorem nfeq 2320

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 2164	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2306	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2306	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1582	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1569	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1467	1 |- F/ x A = B

Syntax hints:   <-> wb 104   A.wal 1346   = wceq 1348   F/wnf 1453   e. wcel 2141   F/_wnfc 2299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301

This theorem is referenced by:  nfel  2321  nfeq1  2322  nfeq2  2324  nfne  2433  raleqf  2661  rexeqf  2662  reueq1f  2663  rmoeq1f  2664  rabeqf  2720  sbceqg  3065  csbhypf  3087  nfiotadw  5163  nffn  5294  nffo  5419  fvmptdf  5583  mpteqb  5586  fvmptf  5588  eqfnfv2f  5597  dff13f  5749  ovmpos  5976  ov2gf  5977  ovmpodxf  5978  ovmpodf  5984  eqerlem  6544  sumeq2  11322  fsumadd  11369  prodeq1f  11515  prodeq2  11520  txcnp  13065  cnmpt11  13077  cnmpt21  13085  cnmptcom  13092