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Theorem nfceqdf 2488
 Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfceqdf.1 xφ
nfceqdf.2 (φA = B)
Assertion
Ref Expression
nfceqdf (φ → (xAxB))

Proof of Theorem nfceqdf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nfceqdf.1 . . . 4 xφ
2 nfceqdf.2 . . . . 5 (φA = B)
32eleq2d 2420 . . . 4 (φ → (y Ay B))
41, 3nfbidf 1774 . . 3 (φ → (Ⅎx y A ↔ Ⅎx y B))
54albidv 1625 . 2 (φ → (yx y Ayx y B))
6 df-nfc 2478 . 2 (xAyx y A)
7 df-nfc 2478 . 2 (xByx y B)
85, 6, 73bitr4g 279 1 (φ → (xAxB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from 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-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2478 This theorem is referenced by:  dfnfc2  3909  nfopd  4605  nfimad  4954  nffvd  5335
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