Theorem nfeqd 2916

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 2730	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
2		nfv 1914	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfeqd.1	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		df-nfc 2892	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
5	3, 4	sylib 218	. . . . 5 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
6	5	19.21bi 2189	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfeqd.2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵)
8		df-nfc 2892	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
9	7, 8	sylib 218	. . . . 5 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
10	9	19.21bi 2189	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
11	6, 10	nfbid 1902	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
12	2, 11	nfald 2328	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
13	1, 12	nfxfrd 1854	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-cleq 2729  df-nfc 2892

This theorem is referenced by:  nfeld  2917  nfeq  2919  nfned  3044  cbvexeqsetf  3495  sbcralt  3872  csbiebt  3928  csbie2df  4443  dfnfc2  4929  eusvnfb  5393  eusv2i  5394  dfid3  5581  iota2df  6548  riotaeqimp  7414  riota5f  7416  oprabid  7463  axrepndlem1  10632  axrepndlem2  10633  axunnd  10636  axpowndlem4  10640  axregndlem2  10643  axinfndlem1  10645  axinfnd  10646  axacndlem4  10650  axacndlem5  10651  axacnd  10652  bj-elgab  36940  bj-gabima  36941  wl-issetft  37583  riotasv2d  38958  nfxnegd  45452