Theorem nfeq 2356

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 2199	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2342	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2342	. . . 4 |- F/ x z e. B
6	3, 5	nfbi 1612	. . 3 |- F/ x ( z e. A <-> z e. B )
7	6	nfal 1599	. 2 |- F/ x A. z ( z e. A <-> z e. B )
8	1, 7	nfxfr 1497	1 |- F/ x A = B

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   F/wnf 1483   e. wcel 2176   F/_wnfc 2335

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337

This theorem is referenced by:  nfel  2357  nfeq1  2358  nfeq2  2360  nfne  2469  raleqf  2698  rexeqf  2699  reueq1f  2700  rmoeq1f  2701  rabeqf  2762  sbceqg  3109  csbhypf  3132  nfiotadw  5235  nffn  5370  nffo  5497  fvmptdf  5667  mpteqb  5670  fvmptf  5672  eqfnfv2f  5681  dff13f  5839  ovmpos  6069  ov2gf  6070  ovmpodxf  6071  ovmpodf  6077  eqerlem  6651  sumeq2  11670  fsumadd  11717  prodeq1f  11863  prodeq2  11868  txcnp  14743  cnmpt11  14755  cnmpt21  14763  cnmptcom  14770  dvmptfsum  15197  lgseisenlem2  15548