Theorem nfeq 2392

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 2226	. 2 |- ( A = B <-> A. z ( z e. A <-> z e. B ) )
2		nfnfc.1	. . . . 5 |- F/_ x A
3	2	nfcri 2378	. . . 4 |- F/ x z e. A
4		nfeq.2	. . . . 5 |- F/_ x B
5	4	nfcri 2378	. . . 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 2203   F/_wnfc 2371

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-cleq 2225  df-clel 2228  df-nfc 2373

This theorem is referenced by:  nfel  2393  nfeq1  2394  nfeq2  2396  nfne  2505  raleqf  2737  rexeqf  2738  reueq1f  2739  rmoeq1f  2740  rabeqf  2803  sbceqg  3154  csbhypf  3177  nfiotadw  5315  nffn  5452  nffo  5589  fvmptdf  5765  mpteqb  5768  fvmptf  5770  eqfnfv2f  5779  dff13f  5943  ovmpos  6177  ov2gf  6178  ovmpodxf  6179  ovmpodf  6185  eqerlem  6798  sumeq2  12044  fsumadd  12092  prodeq1f  12238  prodeq2  12243  txcnp  15136  cnmpt11  15148  cnmpt21  15156  cnmptcom  15163  dvmptfsum  15590  lgseisenlem2  15944