Theorem nfeq 2290

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 2134	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2276	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2276	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1569	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1556	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1451	1 |- F/ x A = B

Syntax hints:   <-> wb 104   A.wal 1330   = wceq 1332   F/wnf 1437   e. wcel 1481   F/_wnfc 2269

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271

This theorem is referenced by:  nfel  2291  nfeq1  2292  nfeq2  2294  nfne  2402  raleqf  2625  rexeqf  2626  reueq1f  2627  rmoeq1f  2628  rabeqf  2679  sbceqg  3023  csbhypf  3043  nfiotadw  5099  nffn  5227  nffo  5352  fvmptdf  5516  mpteqb  5519  fvmptf  5521  eqfnfv2f  5530  dff13f  5679  ovmpos  5902  ov2gf  5903  ovmpodxf  5904  ovmpodf  5910  eqerlem  6468  sumeq2  11160  fsumadd  11207  prodeq1f  11353  prodeq2  11358  txcnp  12479  cnmpt11  12491  cnmpt21  12499  cnmptcom  12506