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Theorem nfcjust 2734
Description: Justification theorem for df-nfc 2735. (Contributed by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
nfcjust (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfcjust
StepHypRef Expression
1 nfv 1828 . . 3 𝑥 𝑦 = 𝑧
2 eleq1 2671 . . 3 (𝑦 = 𝑧 → (𝑦𝐴𝑧𝐴))
31, 2nfbidf 2076 . 2 (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑧𝐴))
43cbvalv 2255 1 (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wal 1472  wnf 1698  wcel 1975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1700  df-cleq 2598  df-clel 2601
This theorem is referenced by: (None)
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