Theorem nfeqd 2504

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 2347	. 2 ⊢ (A = B ↔ ∀y(y ∈ A ↔ y ∈ B))
2		nfv 1619	. . 3 ⊢ Ⅎyφ
3		nfeqd.1	. . . . 5 ⊢ (φ → ℲxA)
4	3	nfcrd 2503	. . . 4 ⊢ (φ → Ⅎx y ∈ A)
5		nfeqd.2	. . . . 5 ⊢ (φ → ℲxB)
6	5	nfcrd 2503	. . . 4 ⊢ (φ → Ⅎx y ∈ B)
7	4, 6	nfbid 1832	. . 3 ⊢ (φ → Ⅎx(y ∈ A ↔ y ∈ B))
8	2, 7	nfald 1852	. 2 ⊢ (φ → Ⅎx∀y(y ∈ A ↔ y ∈ B))
9	1, 8	nfxfrd 1571	1 ⊢ (φ → Ⅎx A = B)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-nfc 2479

This theorem is used by:  nfeld  2505  nfned  2613  vtoclgft  2906  sbcralt  3119  csbiebt  3173  dfnfc2  3910  nfiotad  4343  iota2df  4366  dfid3  4769  oprabid  5551