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Theorem nfcjust 2781
Description: Justification theorem for df-nfc 2782. (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 1883 . . 3 𝑥 𝑦 = 𝑧
2 eleq1 2718 . . 3 (𝑦 = 𝑧 → (𝑦𝐴𝑧𝐴))
31, 2nfbidf 2130 . 2 (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑧𝐴))
43cbvalv 2309 1 (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1521  wnf 1748  wcel 2030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750  df-cleq 2644  df-clel 2647
This theorem is referenced by: (None)
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