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Theorem nfceqdf 2174
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 2104 . . . 4 (φ → (y Ay B))
41, 3nfbidf 1429 . . 3 (φ → (Ⅎx y A ↔ Ⅎx y B))
54albidv 1702 . 2 (φ → (yx y Ayx y B))
6 df-nfc 2164 . 2 (xAyx y A)
7 df-nfc 2164 . 2 (xByx y B)
85, 6, 73bitr4g 212 1 (φ → (xAxB))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164
This theorem is referenced by:  nfopd  3557  dfnfc2  3589  nfimad  4620  nffvd  5130
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