MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfceqdf Structured version   Visualization version   GIF version

Theorem nfceqdf 2975
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 2899 . . . 4 (𝜑 → (𝑦𝐴𝑦𝐵))
41, 3nfbidf 2227 . . 3 (𝜑 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑦𝐵))
54albidv 1921 . 2 (𝜑 → (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑦𝑥 𝑦𝐵))
6 df-nfc 2962 . 2 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
7 df-nfc 2962 . 2 (𝑥𝐵 ↔ ∀𝑦𝑥 𝑦𝐵)
85, 6, 73bitr4g 317 1 (𝜑 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wnf 1785  wcel 2114  wnfc 2960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-cleq 2815  df-clel 2894  df-nfc 2962
This theorem is referenced by:  nfopd  4795  dfnfc2  4835  nfimad  5916  nffvd  6664  riotasv2d  36211  nfcxfrdf  36220  nfded  36221  nfded2  36222  nfunidALT2  36223
  Copyright terms: Public domain W3C validator