Theorem nfeqd 2407

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 2232	. 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
2		nfv 1581	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfeqd.1	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4	3	nfcrd 2406	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
5		nfeqd.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6	5	nfcrd 2406	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵)
7	4, 6	nfbid 1641	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
8	2, 7	nfald 1813	. 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
9	1, 8	nfxfrd 1528	1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ 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-5 1500  ax-7 1501  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-cleq 2231  df-nfc 2381

This theorem is referenced by:  nfeld  2408  nfned  2514  vtoclgft  2873  sbcralt  3128  sbcrext  3129  csbiebt  3187  dfnfc2  3948  eusvnfb  4595  eusv2i  4596  iota2df  5358  riotaeqimp  6053  riota5f  6055