Theorem nfeqd 2905

Description: Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfeqd.1	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfeqd.2	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfeqd	⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Proof of Theorem nfeqd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcleq 2724	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
2		nfv 1915	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfeqd.1	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		df-nfc 2881	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
5	3, 4	sylib 218	. . . . 5 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
6	5	19.21bi 2192	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfeqd.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵)
8		df-nfc 2881	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
9	7, 8	sylib 218	. . . . 5 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
10	9	19.21bi 2192	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
11	6, 10	nfbid 1903	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
12	2, 11	nfald 2329	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
13	1, 12	nfxfrd 1855	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2111  Ⅎwnfc 2879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-cleq 2723  df-nfc 2881

This theorem is referenced by:  nfeld  2906  nfeq  2908  nfned  3030  cbvexeqsetf  3451  sbcralt  3818  csbiebt  3874  csbie2df  4390  dfnfc2  4878  eusvnfb  5329  eusv2i  5330  dfid3  5512  iota2df  6468  riotaeqimp  7329  riota5f  7331  oprabid  7378  axrepndlem1  10483  axrepndlem2  10484  axunnd  10487  axpowndlem4  10491  axregndlem2  10494  axinfndlem1  10496  axinfnd  10497  axacndlem4  10501  axacndlem5  10502  axacnd  10503  bj-elgab  36983  bj-gabima  36984  wl-issetft  37626  riotasv2d  39066  nfxnegd  45549