Theorem nfeq 2400

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

Hypotheses

Ref	Expression
nfnfc.1	⊢ Ⅎ𝑥𝐴
nfeq.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfeq	⊢ Ⅎ𝑥 𝐴 = 𝐵

Proof of Theorem nfeq

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2232	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
2		nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2386	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4		nfeq.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2386	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
6	3, 5	nfbi 1642	. . 3 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)
7	6	nfal 1629	. 2 ⊢ Ⅎ𝑥∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)
8	1, 7	nfxfr 1527	1 ⊢ Ⅎ𝑥 𝐴 = 𝐵

Syntax hints:   ↔ wb 105  ∀wal 1400   = wceq 1402  Ⅎwnf 1513   ∈ wcel 2209  Ⅎwnfc 2379

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381

This theorem is referenced by:  nfel  2401  nfeq1  2402  nfeq2  2404  nfne  2513  raleqf  2745  rexeqf  2746  reueq1f  2747  rmoeq1f  2748  rabeqf  2811  sbceqg  3163  csbhypf  3186  nfiotadw  5335  nffn  5472  nffo  5609  fvmptdf  5787  mpteqb  5790  fvmptf  5792  eqfnfv2f  5801  dff13f  5966  ovmpos  6202  ov2gf  6203  ovmpodxf  6204  ovmpodf  6210  eqerlem  6828  sumeq2  12103  fsumadd  12151  prodeq1f  12297  prodeq2  12302  txcnp  15295  cnmpt11  15307  cnmpt21  15315  cnmptcom  15322  dvmptfsum  15749  lgseisenlem2  16104