Theorem nfeqd 2955

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 2787	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
2		nfv 1890	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfeqd.1	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2939	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfeqd.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6	5	nfcrd 2939	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
7	4, 6	nfbid 1882	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
8	2, 7	nfald 2308	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
9	1, 8	nfxfrd 1833	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1518   = wceq 1520  Ⅎwnf 1763   ∈ wcel 2079  Ⅎwnfc 2931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1760  df-nf 1764  df-cleq 2786  df-nfc 2933

This theorem is referenced by:  nfeld  2956  nfeq  2958  nfned  3086  vtoclgft  3491  vtoclgftOLD  3492  sbcralt  3778  csbiebt  3832  dfnfc2  4757  eusvnfb  5178  eusv2i  5179  dfid3  5341  iota2df  6205  riotaeqimp  6991  riota5f  6993  oprabid  7038  axrepndlem1  9849  axrepndlem2  9850  axunnd  9853  axpowndlem4  9857  axregndlem2  9860  axinfndlem1  9862  axinfnd  9863  axacndlem4  9867  axacndlem5  9868  axacnd  9869  riotasv2d  35574  nfxnegd  41211